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3 Linear Equations and Inequalities in Two Variables

1. 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 147 o REVISED PAGES 3 3 Distance and Slope 147 Po x2 y2 Vertical Vertical change l Pi X1 y1 8 change ae Y2 yp i Ro y Horizontal change Horizontal change 5 x2 x1 Slope Vertical change OP Horizontal change Figure 3 26 Figure 3 27 Because pe E za how we designate P and P is not important Let s use X274 XT 2 Definition 3 1 to find the slopes of some lines EXAMPLE 5 Find the slope of the line determined by each of the following pairs of points and graph the lines a 1 1 and 3 2 b 4 2 and 1 5 c 2 3 and 3 3 Solution a Let 1 1 be P and 3 2 be P Figure 3 28 ya yi 2 1 1 m x27 x 3 1 4 Figure 3 28 b Let 4 2 be P and 1 5 be P Figure 3 29 y2 7 _5 2 _7 7 X3 X 1 4 5 5 m b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 148 p REVISED PAGES 148 Chapter 3 Linear Equations and Inequalities in Two Variables c Let 2 3 be P and 3 3 be P Figure 3 30 Vaya m X27 X1 er dk 3 2 0 0 5 IEE PEE Figure 3 29 Figure 3 30 Y PRACTICE YOUR SKILL Find the slope of the line determined by each of the following pairs of points and graph the lines a 4 2 and 2 5 b 3 4 and 1 4 c 3 2 and 0 2 See answer sectio
2. z 2 4 4 5 5 03 W4928 AM1 qxd 11 3 08 8 27 PM Page 167 o REVISED PAGES 3 4 Determining the Equation of a Line 167 FURTHER INVESTIGATIONS 82 The equation of a line that contains the two points x y1 and x5 y2 is E We often refer wxi X XY to this as the two point form of the equation of a straight line Use the two point form and write the equation of the line that contains each of the indicated pairs of points Express final equations in standard form a 1 1 and 5 2 x 4y 3 b 2 4 and 2 1 5x 4y 6 c 3 5 and 3 1 2x 3y 9 d 5 1 and 2 7 8x 7y 33 83 Let Ax By C and A x B y C represent two lines Change both of these equations to slope intercept form and then verify each of the following properties A B a If x RB 5 then the lines are parallel b If AA BB then the lines are perpendicular 84 The properties in Problem 83 provide us with another way to write the equation of a line parallel or perpendi cular to a given line that contains a given point not on the line For example suppose that we want the equation of the line perpendicular to 3x 4y 6 that contains the point 1 2 The form 4x 3y k where k is a constant represents a family of lines perpendicular to 3x 4y 6 because we have satisfied the condition AA BB Therefore to find what specific line of the family contains 1 2 we substitute
3. 2 5 and parallel to the line x 2y 4 x 2y 8 Containing the point 2 6 and perpendicular to the line 3x 2y 12 2x 3y 14 Containing the point 8 3 and parallel to the line 4x y 7 4x y 29 The taxes for a primary residence can be described by a linear relationship Find the equation for the relationship if the taxes for a home valued at 200 000 are 2400 and the taxes are 3150 when the home is valued at 250 000 Let y be the taxes and x the value of the home Write the equation in slope intercept form See below The freight charged by a trucking firm for a parcel under 200 pounds depends on the miles it is being shipped To ship a 150 pound parcel 300 miles it costs 40 If the same parcel is shipped 1000 miles the cost is 180 Assume the relationship between the cost and miles is linear Find the equation for the relationship Let y be the cost and x be the miles Write the equation in slope intercept form y ix 20 50 y 24 600 200 03 W4928 AM1 qxd 11 3 08 176 52 8 27 PM Page 176 p REVISED PAGES Chapter 3 Linear Equations and Inequalities in Two Variables On a final exam in math class the number of points earned has a linear relationship with the number of cor rect answers John got 96 points when he answered 12 questions correctly Kimberly got 144 points when she answered 18 questions correctly Find the equation for the relationship Let y be the number of points
4. 2x is shown in Figure 3 8 Intercepts Additional point Check point Figure 3 8 Y PRACTICE YOUR SKILL Graph y 3x See answer section EXAMPLE 6 Graph x 2 Solution Because we are considering linear equations in two variables the equation x 2 is equivalent to x O y 2 Now we can see that any value of y can be used but the x value must always be 2 Therefore some of the solutions are 2 0 2 1 2 2 2 1 and 2 2 The graph of all solutions of x 2 is the vertical line in Figure 3 9 Figure 3 9 b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 130 F REVISED PAGES 130 Chapter 3 Linear Equations and Inequalities in Two Variables Y PRACTICE YOUR SKILL Graph x 3 See answer section EXAMPLE 7 Graph y 3 Solution The equation y 3 is equivalent to 0 x y 3 Thus any value of x can be used but the value of y must be 3 Some solutions are 0 3 1 3 2 3 1 3 and 2 3 The graph of y 3 is the horizontal line in Figure 3 10 X v Figure 3 10 Y PRACTICE YOUR SKILL Graph y 4 See answer section E In general the graph of any equation of the form Ax By C where A 0 or B 0 not both is a line parallel to one of the axes More specifically any equation of the form x a where a is a constant is a line parallel to the y axis that has an x intercept of a Any equation of the form y b where b is a constan
5. 4 7y 3 4 and 1 3x 4 7y 11 6 c 2 7x 3 9y 1 4 and 2 7x 3 9y 8 2 d 5x 7y 17 and 7x 5y 19 e 9x 2y 14 and 2x 9y 17 2 1x 3 4y 11 7 and 3 4x 2 1ly 17 3 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 155 F REVISED PAGES 3 3 Distance and Slope 155 1 True 2 True 3 False 4 False 5 True 6 False 7 True 8 False 9 True 10 False 1 8units 2 13 units 3 V29units 4 d d V4 d V82 5 a m gt b m 0 y y 2 5 3 4 1 4 4 2 x x 4 c m a To y y 0 2 2 0 X X CE2 6 y 8 50 ft G2 X 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 156 F REVISED PAGES 156 Chapter 3 Linear Equations and Inequalities in Two Variables 3 4 Determining the Equation of a Line OBJECTIVES Find the Equation of a Line Given a Point and a Slope Find the Equation of a Line Given Two Points Find the Equation of a Line Given the Slope and y Intercept Use the Point Slope Form to Write Equations of Lines Apply the Slope Intercept Form of an Equation cofof f ini Find the Equations for Parallel or Perpendicular Lines El Find the Equation of a Line Given a Point and a Slope To review there are basically two types of problems to solve in coordinate geometry 1 Given an algebraic equation find its geometric graph 2 Given a set of conditions pertaining to a geometric figure find its algebraic equation Problems of type 1 have been our primary concern thus far in this chap
6. 1 for x and 2 for y to determine k 4x 3y k 4 1 3 2 k 2 k Thus the equation of the desired line is 4x 3y 2 Use the properties from Problem 83 to help write the equation of each of the following lines a Contains 1 8 and is parallel to 2x 3y 6 2x 3y 26 b Contains 1 4 and is parallel tox 2y 4 x 2y 9 GRAPHING CALCULATOR ACTIVITIES c Contains 2 7 and is perpendicular to 3x Sy 10 5x 3y 11 d Contains 1 4 and is perpendicular to 2x 5y 12 5x 2y 3 85 The problem of finding the perpendicular bisector of a line segment presents itself often in the study of analytic geometry As with any problem of writing the equation of a line you must determine the slope of the line and a point that the line passes through A perpendicular bisec tor passes through the midpoint of the line segment and has a slope that is the negative reciprocal of the slope of the line segment The problem can be solved as follows Find the perpendicular bisector of the line segment between the points 1 2 and 7 8 1 7 2 8 The midpoint of the line segment is 2 2 a 4 3 8 2 10_5 f 6 3 Hence the perpendicular bisector will pass through the The slope of the line segment is m point 4 3 and have a slope of m 5 3 y 3 Z x 4 5 y 3 3 x 4 Sy 15 3x 12 3x 5y 27 Thus the equation of the perpendicular bisector
7. 2 0 1 and 3 Organize the information into a table E 4 12 2 8 0 4 1 2 3 2 A table can show some of the infinite number of solutions for a linear equation in two variables but for a visual display solutions are plotted on a coordinate system Let s review the rectangular coordinate system and then we can use a graph to display the solutions of an equation in two variables Review of the Rectangular Coordinate System Consider two number lines one vertical and one horizontal perpendicular to each other at the point we associate with zero on both lines Figure 3 1 We refer to these number lines as the horizontal and vertical axes or together as the coordinate axes They partition the plane into four regions called quadrants The quadrants are num bered counterclockwise from I through IV as indicated in Figure 3 1 The point of in tersection of the two axes is called the origin It is now possible to set up a one to one correspondence between ordered pairs of real numbers and the points in a plane To each ordered pair of real numbers there corresponds a unique point in the plane and to each point in the plane there corre sponds a unique ordered pair of real numbers A part of this correspondence is illus tratedin Figure 3 2 The ordered pair 3 2 means that the point A is located three units to the right of and two units up from the origin The ordered pair 0 0 is associated with the origin O The o
8. 3 08 8 26 PM Page 162 F REVISED PAGES 162 Chapter 3 Linear Equations and Inequalities in Two Variables EXAMPLE 8 slopes is 1 Details for verifying these facts are left to another course In other words if two lines have slopes m and m respectively then 1 The two lines are parallel if and only if m m 2 The two lines are perpendicular if and only if m m 1 The following examples demonstrate the use of these properties a Verify that the graphs of 2x 3y 7 and 4x 6y 11 are parallel lines b Verify that the graphs of 8x 12y 3 and 3x 2y 2 are perpendicular lines Solution a Let s change each equation to slope intercept form 2x 3y 7 ae 3y 2x 7 pr ia ier 4x 6y 11 gt 6y 4x 11 4 u y a y 2 n ESE aa 2 Both lines have a slope of T7 but they have different y intercepts There fore the two lines are parallel b Solving each equation for y in terms of x we obtain amp 12y 3 12y 8x 3 8 _ 23 VTR R2 2 1 a 4 3x 2y 2 2y 3x 2 3 Paget 2 3 Because 2 gt 1 the product of the two slopes is 1 the lines are perpendicular Y PRACTICE YOUR SKILL a Verify that the graphs of x 3y 2 and 2x 6y 7 are parallel lines b Verify that the graphs of 2x Sy 3 and 5x 2y 8 are perpendicular i 1 2 5 lines a m m 3 b m 5M gt a Remark The statement the product of t
9. 33 y 2x 1 34 y 3x 4 35 y 0x 4 36 y 2x 0 b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 166 p REVISED PAGES 166 Chapter 3 Linear Equations and Inequalities in Two Variables Use the Point Slope Form to Write Equations of Lines For Problems 37 42 use the point slope form to write the equation of the line that has the indicated slope and contains the indicated point Express the final answer in standard form 37 m gt 3 4 5x 2y 23 2 38 m 7 1 4 2x 3y 14 39 m 2 5 8 2x y 18 40 m 1 6 2 x y 4 1 4 m T7 5 0 x 3y 5 3 42 m F 0 1 3x 4y 4 Apply the Slope Intercept Form of an Equation For Problems 43 48 change the equation to slope intercept form and determine the slope and y intercept of the line gt 43 3x y 7 gt 44 5x y 9 m 3p 7 m 5 0 9 45 3x 2y 9 46 x 4y 3 See below See below 47 x 5y 12 48 4x 7y 14 See below See below For Problems 49 56 use the slope intercept form to graph the following lines See answer section 2 l gt 49 ke W y T2 51 y 2x 1 52 y 3x 1 5 P53 y Sx 4 54 aaa oe 55 y x 2 56 y 2x 4 For Problems 57 66 graph the following lines using the tech nique that seems most appropriate See answer section J 1 57 y x 1 Sh Yeats 5 59 x 2y 5 60 2x y 7 61 y 4x 7 62 3x 2y 63 Ty 2x 64 y 3 65 x 2 66 y x 4 Find the Equations for Parallel or Perpendicular Lines Fo
10. Graph y gt 2x See answer section CONCEPT Q UIZ For Problems 1 10 answer true or false 1 A dashed line on the graph indicates that the points on the line do not satisfy The ordered pair 2 3 satisfies the inequality 2x y gt 1 the inequality Any point can be used as a test point to determine the half plane that is the so lution of the inequality The ordered pair 3 2 satisfies the inequality 5x 2y 19 The ordered pair 1 3 satisfies the inequality 2x 3y lt 4 The graph of x gt 0 is the half plane above the x axis The graph of y lt 0 is the half plane below the x axis The graph of x y gt 4is the half plane above the line x y 4 The origin can serve as a test point to determine the half plane that satisfies the inequality 3y gt 2x The ordered pair 2 1 can be used as a test point to determine the half plane that satisfies the inequality y lt 3x 7 03 W4928 AM1 qxd 142 11 3 08 8 26 PM Page 142 p REVISED PAGES Chapter 3 Linear Equations and Inequalities in Two Variables Problem Set 3 2 El Graph Linear Inequalities For Problems 1 24 graph each of the inequalities See answer section gt 1 gt 3 gt 5 gt 7 9 gt 11 gt 13 x yo gt 2 xty gt 4 x 3y lt 3 4 2x y gt 6 2x 5y 10 gt 6 3x 2y 4 ys x 2 8 y 2x 1 yor x 10 y lt x 2x y 0 gt 12 x 2y 0 x 4y 4
11. The area of a sidewalk whose width is fixed at 3 feet can be given by the equation A 3 where A represents the area in square feet and represents the length in feet Label the horizontal axis and the vertical axis A and graph the equation A 3 for nonnegative values of I See answer section 47 An online grocery store charges for delivery based on the equation C 0 30p where C represents the cost in dollars and p represents the weight of the groceries in pounds Label the horizontal axis p and the vertical axis C and graph the equation C 0 30p for nonnegative values of p See answer section THOUGHTS INTO WORDS 48 How do we know that the graph of y 3x is a straight line that contains the origin gt 49 How do we know that the graphs of 2x 3y 6 and 2x 3y 6 are the same line 50 What is the graph of the conjunction x 2 and y 4 What is the graph of the disjunction x 2 or y 4 Explain your answers 51 Your friend claims that the graph of the equation x 2 is the point 2 0 How do you react to this claim FURTHER INVESTIGATIONS From our work with absolute value we know that x y 1 is equivalent tox y 1 orx y 1 Therefore the graph of x y 1 consists of the two lines x y 1 and x y 1 Graph each of the following See answer section iat This is the first of many appearances of a group of problems called graphing calculator activities These pr
12. answers in radical form gt 1 3 gt 5 7 9 11 gt 13 14 15 16 For 2 1 7 11 18 2 2 1 10 7 10 1 1 3 4 V13 4 1 3 2 2 V34 6 5 9 7 V97 6 4 2 1 6 5 3 3 0 2 V34 8 1 4 4 0 vm 1 6 5 6 6 10 2 3 2 7 10 1 7 4 1 V3 P12 6 4 3 8 V153 Verify that the points 0 2 0 7 and 12 7 are vertices of a right triangle Hint If a b c then it is a right triangle with the right angle opposite side c Verify that the points 0 4 3 0 and 3 0 are vertices of an isosceles triangle Verify that the points 3 5 and 5 8 divide the line seg ment joining 1 2 and 7 11 into three segments of equal length Verify that 5 1 is the midpoint of the line segment joining 2 6 and 8 4 Find the Slope of a Line Problems 17 28 graph the line determined by the two points and find the slope of the line P17 19 gt 21 1 2 4 6 18 3 1 2 2 2 4 5 1 2 2 20 2 5 3 1 2 6 6 2 2 22 2 1 2 5 1 gt Blue arrows indicate Enhanced WebAssign problems 23 25 27 gt 29 30 31 gt 32 6 1 1 4 24 3 3 2 3 0 2 4 2 4 0 26 1 5 4 1 4 0 2 4 0 gt 28 4 0 0 6 gt Find x if the line through 2 4 and x 6 has a slope 2 f F g Find y if th
13. are of the form Ax By gt C or Ax By lt C where A B and C are real numbers Combined linear equality and inequality state ments are of the form Ax By C or Ax By SC Graphing linear inequalities is almost as easy as graphing linear equations The following discussion leads into a simple step by step process Let s consider the fol lowing equation and related inequalities xty 2 xt ty gt 2 xty lt 2 The graph of x y 2 is shown in Figure 3 16 The line divides the plane into two half planes one above the line and one below the line In Figure 3 17 a we have indicated 0 2 2 0 xX Figure 3 16 a b Figure 3 17 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 139 p REVISED PAGES 3 2 Linear Inequalities in Two Variables 139 several points in the half plane above the line Note that for each point the ordered pair of real numbers satisfies the inequality x y gt 2 This is true for all points in the half plane above the line Therefore the graph of x y gt 2 is the half plane above the line as indicated by the shaded portion in Figure 3 17 b We use a dashed line to indicate that points on the line do not satisfy x y gt 2 We would use a solid line if we were graphing x y 2 In Figure 3 18 a several points were indicated in the half plane below the line x y 2 Note that for each point the ordered pair of real numbers satisfies the in equality x y lt 2 This is true for all
14. horizontal axis as the x axis and the vertical axis as the y axis as in Figure 3 4 a Connecting the points with a straight line as in Figure 3 4 b produces a graph of the equation y x 2 Every point on the line has coordinates that are solutions of the equation y x 2 The graph provides a visual display of all the infinite solutions for the equation a b Figure 3 4 EXAMPLE 2 Graph the equation y x 4 Solution Let s begin by determining some solutions for the equation y x 4 and then plot the solutions on a rectangular coordinate system to produce a graph of the equation Let s use a table to record some of the solutions 4 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 126 p REVISED PAGES 126 Chapter 3 Linear Equations and Inequalities in Two Variables y value determined x value from y x 4 Ordered pairs 3 7 1 5 0 4 2 2 4 0 2 6 2 O N AUNI We can plot the ordered pairs on a coordinate system as shown in Figure 3 5 a The graph of the equation is produced by drawing a straight line through the plotted points as in Figure 3 5 b Figure 3 5 Y PRACTICE YOUR SKILL Graph the equation y 2x 2 See answer section Oo Graph Linear Equations by Finding the x and y Intercepts The points 4 0 and 0 4 in Figure 3 5 b are special points They are the points of the graph that are on the coordinate axes That is they yield the x intercept and the
15. is gt 21 y xt3 22 y x 1 the hemoglobin Alc reading and G is the average gt 23 y 2x 1 24 y 4x 3 blood glucose reading Complete this chart of values 2 3 gt 25 y 5x 3 26 y so Hemoglobin Alc h 6 0 6 5 7 0 80 85 9 0 10 0 gt 27 3y x 3 28 2yv x 2 Blood glucose G 120 135 150 180 195 210 240 gt Blue arrows indicate Enhanced WebAssign problems b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 135 3 1 b Label the horizontal axis h and the vertical axis G then graph the equation G 30h 60 for h values between 4 0 and 12 0 See answer section c Use the graph from part b to approximate values for G when h 5 5 and 7 5 d Check the accuracy of your readings from the graph in part c by using the equation G 30h 60 G 105 G 165 44 Suppose that the daily profit from an ice cream stand is given by the equation p 2n 4 where n represents the gallons of ice cream mix used in a day and p represents the dollars of profit Label the horizontal axis n and the verti cal axis p and graph the equation p 2n 4 for nonneg ative values of n See answer section gt 45 The cost c of playing an online computer game for a time t in hours is given by the equation c 3t 5 Label the p REVISED PAGES Rectangular Coordinate System and Linear Equations 135 horizontal axis and the vertical axis c and graph the equa tion for nonnegative values of t See answer section 46
16. of the line segment between the points 1 2 and 7 8 is 3x 5y 27 Find the perpendicular bisector of the line segment be tween the points for the following Write the equation in standard form a 1 2 and 3 0 2x y 1 b 6 10 and 4 2 5x 6y 29 c 7 3 and 5 9 x y 2 d 0 4 and 12 4 3x 2y 18 86 Predict whether each of the following pairs of equations represents parallel lines perpendicular lines or lines that intersect but are not perpendicular Then graph each pair of lines to check your predictions The properties pre sented in Problem 83 should be very helpful a 5 2x 3 3y 9 4 and 5 2x 3 3y 12 6 b 1 3x 4 7y 3 4 and 1 3x 4 7y 11 6 c 2 7x 3 9y 1 4 and 2 7x 3 9y 8 2 d 5x 7y 17 and 7x 5y 19 e 9x 2y 14 and 2x 9y 17 2 1x 3 4y 11 7 and 3 4x 2 1y 17 3 g 7 1x 2 3y 6 2 and 2 3x 7 1y 9 9 h 3x 9y 12 and 9x 3y 14 i 2 6x 5 3y 3 4 and 5 2x 10 6y 19 2 j 4 8x 5 6y 3 4 and 6 1x 7 6y 12 3 03 W4928 AM1 qxd 11 3 08 8 27 PM Page 168 P REVISED PAGES 168 Chapter 3 Linear Equations and Inequalities in Two Variables 03 W4928 AM1 qxd 11 3 08 8 27 PM Page 169 o REVISED PAGES OBJECTIVE SUMMARY EXAMPLE CHAPTER REVIEW PROBLEMS Find solutions for lin ear equations in two variables Sec 3 1 Obj 1 p 122
17. p 161 _ 5 only if m m Solution 2 The two lines are perpendicular The slope of the parallel line is 3 if and only if m m 1 Therefore use this slope and the point 2 1 to determine the equation y 1 3 x 2 Simplifying this equation yields y 3x 5 Chapter 3 Review Problem Set For Problems 1 4 determine which of the ordered pairs are 3x 4 solutions of the given equation 7 y E 1 4 y 6 1 2 6 0 1 10 1 2 1 10 2 x 2y 4 4 1 4 1 0 2 0 2 8 amaya x 0 3 3 3x 2y 12 2 3 2 9 3 2 2 3 2 9 y a lt i 9 4 2x 3y 6 0 2 3 0 1 2 3 0 For Problems 9 12 graph each equation by finding the x and y intercepts See answer section For Problems 5 8 complete the table of values for the equa 9 2x y 6 ii ey 6 tion and graph the equation See answer section 11 x 2y 4 12 5x y 5 5 y 2x 5 P 1 01 4 y y y 7 gt 3 3 For Problems 13 18 graph each equation See answer section 13 y 4x 14 2x 3y 0 6 y 2x 1 P 3 1 0 2 15 x 1 16 y 2 17 y 4 18 x 3 b 03 W4928 AM1 qxd 19 20 11 3 08 8 27 PM Page 175 a An apartment moving company charges according to the equation c 75h 150 where c represents the charge in dollars and h represents the number of hours for the move Complete the following table h 1 2 3 4 c 225 300 375 450 b Labeling the horizontal axis h and the ver
18. points in the half plane below the line Thus the graph of x y lt 2 is the half plane below the line as indicated in Figure 3 18 b a b Figure 3 18 To graph a linear inequality we suggest the following steps 1 First graph the corresponding equality Use a solid line if equality is included in the original statement use a dashed line if equality is not included Choose a test point not on the line and substitute its coordinates into the inequality The origin is a convenient point to use if it is not on the line The graph of the original inequality is a the half plane that contains the test point if the inequality is satisfied by that point or b the half plane that does not contain the test point if the inequality is not satisfied by the point Let s apply these steps to some examples EXAMPLE 1 Graph x 2y gt 4 Solution Step 1 Step 2 Step 3 Graph x 2y 4 as a dashed line because equality is not included in x 2y gt 4 Figure 3 19 Choose the origin as a test point and substitute its coordinates into the inequality x 2y gt 4 becomes 0 2 0 gt 4 which is false Because the test point did not satisfy the given inequality the graph is the half plane that does not contain the test point Thus the graph of x 2y gt 4 is the half plane below the line as indicated in Figure 3 19 4 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 140 F REVISED PAGES 140 C
19. slope of TT then by starting at some point on the line we could locate other points on the line as follows 3 G a by moving 3 units down and 4 units to the right 3 3 a oF gt by moving 3 units up and 4 units to the left 3 9 G by moving 9 units down and 12 units to the right 3 15 4 0 by moving 15 units up and 20 units to the left Use Slope to Graph Lines 1 EXAMPLE 6 Graph the line that passes through the point 0 2 and has a slope of 3 Solution To graph plot the point 0 2 Furthermore because the slope is equal to vertical change 1 i i we can locate another point on the line by starting from horizontal change 3 the point 0 2 and moving 1 unit up and 3 units to the right to obtain the point 3 1 Because two points determine a line we can draw the line Figure 3 31 Figure 3 31 j Remark Because m 3 Ty we can locate another point by moving 1 unit down and 3 units to the left from the point 0 2 V PRACTICE YOUR SKILL 2 Graph the line that passes through the point 3 2 and has a slope of 5 See answer section b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 150 F REVISED PAGES 150 Chapter 3 Linear Equations and Inequalities in Two Variables EXAMPLE 7 Graph the line that passes through the point 1 3 and has a slope of 2 Solution 2 To graph the line plot the point 1 3 We know that m 2 41 Furthermore
20. straight line there is no need for any extensive table of values Furthermore the solution 2 3 served as a check point If it had not been on the line determined by the two intercepts then we would have known that an error had been made EXAMPLE 4 Graph 2x 3y 7 Solution Without showing all of our work the following table indicates the intercepts and a check point The points from the table are plotted and the graph of 2x 3y 7 is shown in Figure 3 7 y intercept Check point Intercepts Check point x intercept Figure 3 7 Y PRACTICE YOUR SKILL Graph x 2y 3 See answer section E b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 129 F REVISED PAGES 3 1 Rectangular Coordinate System and Linear Equations 129 Graph Lines Passing through the Origin Vertical Lines and Horizontal Lines It is helpful to recognize some special straight lines For example the graph of any equation of the form Ax By C where C 0 the constant term is zero is a straight line that contains the origin Let s consider an example EXAMPLE 5 Graph y 2x Solution Obviously 0 0 is a solution Also notice that y 2x is equivalent to 2x y 0 thus it fits the condition Ax By C where C 0 Because both the x intercept and the y intercept are determined by the point 0 0 another point is necessary to deter mine the line Then a third point should be found as a check point The graph of y
21. the problem However sometimes it is helpful to use a figure to organize the given information and aid in analyzing the problem as we see in the next example EXAMPLE 4 Verify that the points 2 2 11 7 and 4 9 are vertices of an isosceles triangle An isosceles triangle has two sides of the same length Solution Let s plot the points and draw the triangle Figure 3 25 Use the distance formula to find the lengths d d and d3 as follows EEBEER d V 4 2 9 2 V2 7P V4 49 V53 d V 11 4 7 972 V49 4 V53 d V 11 2 7 27 V 5 Figure 3 25 V81 25 V106 Because d d we know that it is an isosceles triangle Y PRACTICE YOUR SKILL Verify that the points 2 2 7 1 and 2 3 are vertices of an isosceles triangle d d VM d V82 E Find the Slope of a Line In coordinate geometry the concept of slope is used to describe the steepness of lines The slope of a line is the ratio of the vertical change to the horizontal change as we move from one point on a line to another point This is illustrated in Figure 3 26 with points P and P A precise definition for slope can be given by considering the coordinates of the points P4 P and R as indicated in Figure 3 27 The horizontal change as we move from P to P is x x and the vertical change is y y Thus the following defini tion for slope is given
22. vertical change because the slope we can locate another point on the horizontal change 1 line by starting from the point 1 3 and moving 2 units down and 1 unit to the right to obtain the point 2 1 Because two points determine a line we can draw the line Figure 3 32 1 3 2 1 Figure 3 32 2 2 Remark Because m 2 1 7 Zq We can locate another point by moving 2 units up and 1 unit to the left from the point 1 3 V PRACTICE YOUR SKILL Graph the line that passes through the point 2 0 and has a slope of m gt See answer section Apply Slope to Solve Problems The concept of slope has many real world applications even though the word slope is often not used The concept of slope is used in most situations where an incline is involved Hospital beds are hinged in the middle so that both the head end and the foot end can be raised or lowered that is the slope of either end of the bed can be changed Likewise treadmills are designed so that the incline slope of the platform can be adjusted A roofer when making an estimate to replace a roof is concerned not only about the total area to be covered but also about the pitch of the roof Con tractors do not define pitch as identical with the mathematical definition of slope but both concepts refer to steepness In Figure 3 33 the two roofs might require the same amount of shingles but the roof on the left will take longer to com
23. 0 14 2x y 3 20 gt 15 gt 17 gt 19 gt 21 gt 23 gt 24 3 Le 16 2x 5y gt 4 Sag yaler y 3 y 3 23 gt 20 y 2 x gt 1 and y lt 3 x gt 2 and y gt 1 xs 1 and y lt 1 x lt 2 and y 2 A THOUGHTS INTO WORDS 25 Why is the point 4 1 not a good test point to use when graphing 5x 2y gt 22 26 Explain how you would graph the 3 gt x Jy inequality FURTHER INVESTIGATIONS gt 27 gt 28 31 Graph x lt 2 Hint Remember that x lt 2 is equivalent to 2 lt x lt 2 See answer section Graph y gt 1 See answer section gt 29 gt 30 GRAPHING CALCULATOR ACTIVITIES See answer section Graph x y lt 1 See answer section Graph x y gt 2 This is a good time for you to become acquainted with the DRAW features of your graphing calculator Again you may need to consult your user s manual for specific key punching instructions Return to Examples 1 2 and 3 of this section and use your graphing calculator to graph the inequalities Use a graphing calculator to check your graphs for Problems 1 24 gt Blue arrows indicate Enhanced WebAssign problems 33 Use the DRAW feature of your graphing calculator to draw each of the following a A line segment between 2 4 and 2 5 b A line segment between 2 2 and 5 2 c A line segment between 2 3 and 5 7 d A triangl
24. 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 121 F REVISED PAGES Linear Equations and Inequalities in Two Variables Leonard de Selva CORBIS E Ren Descartes a philosopher and mathematician developed a system for locating a point on a plane This system is our current rectangular coordinate grid used for graphing it is named the Cartesian coordinate system R en Descartes a French mathematician of the 17th century was able to trans form geometric problems into an algebraic setting so that he could use the tools of algebra to solve the problems This connecting of algebraic and geometric ideas is the foundation of a branch of mathematics called analytic geometry today more commonly called coordinate geometry Basically there are two kinds of prob lems in coordinate geometry Given an algebraic equation find its geometric graph and given a set of conditions pertaining to a geometric graph find its algebraic equa tion We discuss problems of both types in this chapter Video tutorials for all section learning objectives are available in a variety of delivery modes b 3 1 Rectangular Coordinate System and Linear Equations 3 2 Linear Inequalities in Two Variables 3 3 Distance and Slope 3 4 Determining the Equation of a Line 121 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 122 F REVISED PAGES INTERNET PROJECT In this chapter the rectangular coordinate system is used for graphing Another two dimensional co
25. 2 Find the distance between the points A 2 3 and B 5 7 Solution Let s plot the points and form a right triangle as indicated in Figure 3 23 Note that the coordinates of point C are 5 3 Because AC is parallel to the horizontal axis its length is easily determined to be 3 units Likewise CB is parallel to the vertical axis and its length is 4 units Let d represent the length of AB and apply the Pythagorean theorem to obtain d 3 4 0 7 BOS 7 d2 9 4 16 4 units d 25 V25 5 Distance between is a nonnegative value so the length of AB is 5 units 2 0 5 0 Figure 3 23 Y PRACTICE YOUR SKILL Find the distance between the points A 4 1 and B 8 6 13 units b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 145 F REVISED PAGES 3 3 Distance and Slope 145 We can use the approach we used in Example 2 to develop a general distance formula for finding the distance between any two points in a coordinate plane The de velopment proceeds as follows 1 Let P x y1 and Pz x2 y2 represent any two points in a coordinate plane 2 Form a right triangle as indicated in Figure 3 24 The coordinates of the vertex of the right angle point R are x2 y1 5 est al aa P22 y2 Pix yy i b2 sil R x y x2 x l I 2 gt V1 Ga 0 X2 0 Figure 3 24 The length of P R is x x and the length of RP is y yj Again the absolut
26. 25 00 6 y 2x 1 y 3s S 3 b Label the horizontal axis m and the vertical axis c and graph the equation c 0 25m 10 for nonnega X 2 0 2 4 tive values of m T RYO c Use the graph from part b to approximate values y Se e for c when m 25 40 and 45 d Check the accuracy of your readings from the graph A ae ete ae in part c by using the equation c 0 25m 10 8 2x 3y 6 9 y M 10 A P42 a The equation F C 32 can be used to convert 3 from degrees Celsius to degrees Fahrenheit Com plete the following table Graph Linear Equations by Finding the x and y Intercepts C O 5 10 15 20 5 10 15 20 25 F 32 41 50 59 68 23 14 5 f For Problems 9 28 graph each of the linear equations by find ing the x and y intercepts b Graph the equation F sc 32 See answer section gt 9 x 2y 4 10 2x y 6 c Use your graph from part b to approximate values _ for F when C 25 30 30 and 40 1 26 y 2 12 3x y 3 d Check the accuracy of your readings from the graph PL3 Seray o 14 2x 3y 6 in part c by using the equation F 2c 32 P15 5x 4y 20 16 4x 3y 12 77 86 22 40 _ E 43 a A doctor s office wants to chart and graph the linear Ee eae 18 5x y 2 relationship between the hemoglobin Alc reading gt 19 x 2y 3 20 3x 2y 12 and the average blood glucose level The equation G 30h 60 describes the relationship where h
27. A solution of an equation in two variables is an ordered pair of real numbers that satisfies the equation Find a solution for the equation 2x 3y 6 Solution Choose an arbitrary value for x and determine the corresponding y value Let x 3 then substi tute 3 for x in the equation 2 3 3y 6 6 3y 6 3y 12 y 4 Therefore the ordered pair 3 4 is a solution Problems 1 4 Graph the solutions for linear equations Sec 3 1 Obj 3 p 125 A graph provides a visual display of all the infinite solutions of an equa tion in two variables The ordered pair solutions for a linear equation can be plotted as points on a rectan gular coordinate system Connect ing the points with a straight line produces a graph of the equation Graph y 2x 3 Solution Find at least three ordered pair solutions for the equation We can determine that 1 5 0 3 and 1 1 are solu tions The graph is shown below Problems 5 8 continued 169 03 W4928 AM1 qxd 11 3 08 8 27 PM Page 170 p REVISED PAGES substitute 0 for x in the equation and then solve for y Plot the inter cepts and connect them with a straight line to produce the graph 170 Chapter 3 Linear Equations and Inequalities in Two Variables CHAPTER REVIEW OBJECTIVE SUMMARY EXAMPLE PROBLEMS Graph linear equations The x intercept is the x coordin
28. MPLE CHAPTER REVIEW PROBLEMS Find the distance between two points Sec 3 3 Obj 1 p 143 The distance between any two points x1 y1 and x y2 is given by the distance formula a Vix x a yi Find the distance between 1 5 and 4 2 Solution d V E x y gt y d V 4 1 2 5 d VBF OF d V9 49 V58 Problems 27 29 Find the slope of a line Sec 3 3 Obj 2 p 146 The slope denoted by m of a line determined by the points x1 y1 and x2 y2 is given by the slope y2 Yi X X where x x formula m Find the slope of a line that contains the points 1 2 and 7 8 Solution Use the slope formula 8 2 6 3 7 1 8 4 m Thus the slope of the line is Problems 30 32 Use slope to graph lines Sec 3 3 Obj 3 p 149 A line can be graphed knowing a point on the line and the slope by plotting the point and from that point using the slope to locate an other point on the line Then those two points can be connected with a straight line to produce the graph Graph the line that contains the point 3 2 and has a slope a Solution From the point 3 2 locate another point by moving up 5 units and to the right 2 units to obtain the point 1 3 Then draw the line Problems 33 36 continued 03 W4928 AM1 qxd 11 3 08 8 27 PM Pag
29. YOUR SKILL Find the equation of the line that contains 2 5 and 4 10 5x 6y 40 a Find the Equation of a Line Given the Slope and y Intercept 1 EXAMPLE 3 Find the equation of the line that has a slope of 4 and a y intercept of 2 Solution A y intercept of 2 means that the point 0 2 is on the line Figure 3 41 Figure 3 41 b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 158 F REVISED PAGES 158 Chapter 3 Linear Equations and Inequalities in Two Variables Choose a variable point x y and proceed as in the previous examples y 2_1 x 0 4 1 x 0 4 y 2 x 4y 8 x 4y 8 V PRACTICE YOUR SKILL Find the equation of the line that has a slope of 5 and a y intercept of 4 a 3x 2y 8 Perhaps it would be helpful to pause a moment and look back over Examples 1 2 and 3 Note that we used the same basic approach in all three situations We chose a variable point x y and used it to determine the equation that satisfies the condi tions given in the problem The approach we took in the previous examples can be generalized to produce some special forms of equations of straight lines Use the Point Slope Form to Write Equations of Lines Generalizing from the previous examples let s find the equation of a line that has a slope of m and contains the point x y1 To use the slope formula we will need two points Choosing a point x y to represent any other point on the line Figure 3 42 and using th
30. an use the point slope form y y m x x or m 2 a The result can be x xX expressed in standard form or slope intercept form Find the equation of a line that contains the point 1 4 and has a slope of Solution Substitute for m and 1 4 for x y1 into the formula y y m x Xi 3 _ 7 4 2 x 1 Simplifying this equation yields 3x 2y 11 Problems 42 44 continued 03 W4928 AM1 qxd 11 3 08 8 27 PM Page 174 F REVISED PAGES 174 Chapter 3 Linear Equations and Inequalities in Two Variables CHAPTER REVIEW OBJECTIVE SUMMARY EXAMPLE PROBLEMS Find the equation ofa First calculate the slope of the line Find the equation of a line that Problems 45 46 line given two points Substitute the slope and the coordi contains the points 3 4 and 50 53 contained in the line nates of one of the points into 6 10 Sec 3 4 Obj 2 p 157 y y m x x orm i First calculate the slope 10 4 6 2 m a 8 Now substitute 2 for m and 3 4 for x y in the formula y y mx x y 4 2 x 3 Simplifying this equation yields 2x y 2 Find the equations for If two lines have slopes m and m Find the equation of a line that Problems 47 49 parallel and perpendi respectively then contains the point 2 1 and is cular lines fe Theiwolinssare parailalifand parallel to the line y 3x 4 Sec 3 4 Obj 6
31. and x be the number of correct answers Write the equation in slope intercept form y 8x 53 The time needed to install computer cables has a linear relationship with the number of feet of cable being installed It takes 1 hours to install 300 feet and 1050 feet can be installed in 4 hours Find the equation for the relationship Let y be the feet of cable installed and x be the time in hours Write the equation in slope intercept form y 300x 150 03 W4928 AM1 qxd 11 3 08 8 27 PM Page 177 REVISED PAGES 1 Determine which of the ordered pairs are solutions of the equation 2x y 6 1 4 2 2 4 2 3 0 10 26 1 4 3 0 10 26 2 Find the slope of the line determined by the points 2 4 and 3 2 3 Find the slope of the line determined by the equation 3x 7y 12 m 2 4 Find the length of the line segment whose endpoints are 4 2 and 3 1 V58 5 What is the slope of all lines that are parallel to the line 7x 2y 9 a 6 What is the slope of all lines that are perpendicular to the line 4x 9y 6 7 The grade of a highway up a hill is 25 How much change in horizontal distance is there if the vertical height of the hill is 120 feet 480 ft 8 Suppose that a highway rises 200 feet in a horizontal distance of 3000 feet Ex press the grade of the highway to the nearest tenth of a percent 6 7 3 PARE 9 If the ratio of rise to run is to
32. aph 2x 3y 4 2x 3y 6 4x 6y 7 and 8x 12y 1 on the same set of axes b Graph 5x 2y 4 5x 2y 3 10x 4y 3 and 15x 6y 30 on the same set of axes c Graph x 4y 8 2x 8y 3 x 4y 6 and 3x 12y 10 on the same set of axes d Graph 3x 4y 6 3x 4y 10 6x 8y 20 and 6x 8y 24 on the same set of axes 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 136 136 e For each of the following pairs of lines a predict whether they are parallel lines and b graph each pair of lines to check your prediction 1 5x 2y 10 and 5x 2y 4 2 x y 6 and x y 4 3 2x y 8 and 4x 2y 2 4 y 0 2x 1 and y 0 2x 4 5 3x 2y 4 and 3x 2y 4 6 4x 3y 8 and 8x 6y 3 7 2x y 10 and 6x 3y 6 8 x 2y 6 and 3x 6y 6 59 Now let s use a graphing calculator to get a graph of 5 i C gF 32 By letting F x and C y we obtain Figure 3 15 Pay special attention to the boundaries on x These values were chosen so that the fraction Maximum value of x minus Minimum value of x 95 would be equal to 1 The viewing window of the graphing calculator used to produce Figure 3 15 is 95 pixels dots 35 10 85 25 Figure 3 15 p REVISED PAGES Chapter 3 Linear Equations and Inequalities in Two Variables wide Therefore we use 95 as the denominator of the fraction We chose the boundaries for y to make sure that the curso
33. ate Graph x 2y 4 Problems 9 12 by finding the x and y of the point where the graph inter i Solution intercepts sects the x axis The y intercept is Let y 0 Sec 3 1 Obj 4 p 126 the y coordinate of the point where uy ree 0 4 the graph intersects the y axis ca B 4 To find the x intercept substitute Ag Let x 0 0 for y in the equation and then 0 2y 4 solve for x To find the y intercept i 7 9 Graph lines passing through the origin vertical lines and hori zontal lines Sec 3 1 Obj 5 p 129 The graph of any equation of the form Ax By C where C 0 is a straight line that passes through the origin Any equation of the form x a where a is a constant is a vertical line Any equation of the form y b where b is a constant is a horizon tal line Graph 3x 2y 0 Solution The equation indicates that the graph will be a line passing through the origin Solving the equation for y gives us 3 y aoa Find at least three ordered pair solutions for the equation We can determine that 2 3 0 0 and 2 3 are solutions The graph is shown below Problems 13 18 continued 03 W4928 AM1 qxd 11 3 08 8 27 PM Page 171 p REVISED PAGES Chapter 3 Summary 171 CHAPTER REVIEW OBJECTIVE SUMMARY EXAMPLE PROBLEMS Apply graphing to lin Many relationships between two Let c represent the cost in Problems 19 20 ear
34. be 4 for the steps of a staircase and the rise is 32 centimeters find the run to the nearest centimeter 43 cm 10 Find the x intercept of the line 3x y 6 2 2 11 Find the y intercept of the line y ot T7 1 12 Graph the line that contains the point 2 3 and has a slope of r See answer section 13 Find the x and y intercepts for the line x 4y 4 and graph the line See answer section For Problems 14 18 graph each equation See answer section 14 y x 3 15 3x y 5 16 3y 2x 17 y 3 x i 18 y 8 y 4 For Problems 19 and 20 graph each inequality See answer section 19 2x y lt 4 20 3x 2y 6 3 21 Find the equation of the line that has a slope of and contains the point 4 5 Express the equation in standard form 3x 2y 2 22 Find the equation of the line that contains the points 4 2 and 2 1 Express the equation in slope intercept form y o b 10 11 13 14 16 17 18 19 20 21 22 177 03 W4928 AM1 qxd 11 3 08 8 27 PM Page 178 F REVISED PAGES Chapter 3 Linear Equations and Inequalities in Two Variables 178 23 23 24 24 25 25 Find the equation of the line that is parallel to the line 5x 2y 7 and contains the point 2 4 Express the equation in standard form 5x 2y 18 Find the equation of t
35. ch chosen value of x determining a corresponding value for y Let s use a table to record some of the solu tions for y 3x 2 y value determined x value from y 3x 2 Ordered pairs 3 3 7 1 1 1 1 0 0 2 1 5 1 5 2 8 2 8 4 14 4 14 122 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 123 p REVISED PAGES 3 1 Rectangular Coordinate System and Linear Equations 123 EXAMPLE 1 Determine some ordered pair solutions for the equation y 2x 5 and record the It Figure 3 1 values in a table Solution We can start by arbitrarily choosing values for x and then determine the corre sponding y value Even though you can arbitrarily choose values for x it is good prac tice to choose some negative values zero and some positive values Let x 4 then according to our equation y 2 4 5 13 Let x 1 then according to our equation y 2 1 5 7 Let x 0 then according to our equation y 2 0 5 5 Let x 2 then according to our equation y 2 2 5 1 Let x 4 then according to our equation y 2 4 5 3 Organizing this information in a chart gives the following table y value determined x value from y 2x 5 Ordered pair 4 13 4 13 1 7 1 7 0 5 0 5 2 1 2 1 3 4 3 V PRACTICE YOUR SKILL Determine the ordered pair solutions for the equation y 2x 4 for the x values of 4
36. d by the equations x 3y 4 and x 3y 4 are perpendi cular lines 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 165 o REVISED PAGES 3 4 Determining the Equation of a Line 165 Problem Set 3 4 El Find the Equation of a Line Given a Point and a Slope For Problems 1 8 write the equation of the line that has the indicated slope and contains the indicated point Express final equations in standard form 1 5 gt 1 m z 3 5 2 m z 2 3 x 2y 7 x 3y 7 gt 3 m 3 2 4 4 m 2 1 6 3x y 10 2x y 4 3 3 7153 2 4 5 m 4 gt 6 m 5 gt 3x 4y 15 aX S 26 3 7 m 4 4 2 8 mas 8 2 5x 4y 28 3x 2y 28 5 gt 9 x intercept of 3 and slope of E 5x By 15 3 10 x intercept of 5 and slope of a0 3x 10y 15 Find the Equation of a Line Given Two Points For Problems 11 22 write the equation of the line that con tains the indicated pair of points Express final equations in standard form gt 11 2 1 6 5 12 1 2 2 5 x y 1 x y 3 13 2 3 2 7 14 3 4 1 2 5x 2y 4 3x 2y 1 15 3 2 4 1 P16 2 5 3 3 x 7y 11 8x 5y 9 17 1 4 3 6 18 3 8 7 2 x 2y 9 3x 2y 25 19 0 0 5 7 gt 20 0 0 S 9 7x 5y 0 9x 5y 0 P21 x intercept of 2 and y intercept of 4 2x y 4 22 x intercept of 1 and y intercept of F
37. d the rectangular coordi nate system or the Cartesian coordinate system a b Figure 3 3 Historically the rectangular coordinate system provided the basis for the development of the branch of mathematics called analytic geometry or what we presently refer to as coordinate geometry In this discipline Ren Descartes a French 17th century mathematician was able to transform geometric problems into an alge braic setting and then use the tools of algebra to solve the problems Basically there are two kinds of problems to solve in coordinate geometry b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 125 F REVISED PAGES 3 1 Rectangular Coordinate System and Linear Equations 125 1 Given an algebraic equation find its geometric graph 2 Given a set of conditions pertaining to a geometric figure find its algebraic equation In this chapter we will discuss problems of both types Let s start by finding the graph of an algebraic equation Graph the Solutions for Linear Equations Let s begin by determining some solutions for the equation y x 2 and then plot the solutions on a rectangular coordinate system to produce a graph of the equation Let s use a table to record some of the solutions Determine y Solutions for Choose x from y x 2 y x 2 0 2 0 2 1 3 1 3 3 5 3 5 5 7 5 7 2 0 2 0 4 2 4 2 6 4 6 4 We can plot the ordered pairs as points in a coordinate system and use the
38. e value is used to ensure a nonnegative value Let d represent the length of P P and apply the Pythagorean theorem to obtain d x xP y2 yif Because a a the distance formula can be stated as a V x x yp wy It makes no difference which point you call P or P when using the distance formula If you forget the formula don t panic Just form a right triangle and apply the Py thagorean theorem as we did in Example 2 Let s consider an example that demon strates the use of the distance formula Answers to the distance problems can be left in square root form or approxi mated using a calculator Radical answers in this chapter will be restricted to radicals that are perfect squares or radicals that do not need to be simplified The skill of sim plifying radicals is covered in Chapter 7 after which you will be able to simplify the answers for distance problems EXAMPLE 3 Find the distance between 1 5 and 1 2 Solution Let 1 5 be P and 1 2 be P Using the distance formula we obtain d V1 EDP 2 5F V2 3 Va 9 VB The distance between the two points is V13 units b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 146 F REVISED PAGES 146 Chapter 3 Linear Equations and Inequalities in Two Variables WY PRACTICE YOUR SKILL Find the distance between the points A 1 3 and B 4 5 V29 units a In Example 3 we did not sketch a figure because of the simplicity of
39. e 173 p REVISED PAGES Chapter 3 Summary 173 CHAPTER REVIEW OBJECTIVE SUMMARY EXAMPLE PROBLEMS Apply slope to solve The concept of slope is used in most A certain highway has a grade of Problems 37 38 problems situations where an incline is in 2 How many feet does it rise Sec 3 3 Obj 4 p 150 volved In highway construction the word grade is used for slope in a horizontal distance of one third of a mile 1760 feet Solution A 2 grade is equivalent to a 2 slope of T00 We can set up the 2 y proportion 100 1760 then solving for y gives us y 35 2 So the highway rises 35 2 feet in one third of a mile Apply the slope intercept form of an equation of a line Sec 3 4 Obj 5 p 159 The equation y mx b is re ferred to as the slope intercept form of the equation of a line If the equation of a nonvertical line is written in this form then the coeffi cient of x is the slope and the con stant term is the y intercept Change the equation 2x 7y 21 to slope intercept form and determine the slope and y intercept Solution Solve the equation 2x 7y 21 for y 2x 7y 21 Ty 2x 21 2 a 2 The slope is 3 and the y inter cept is 3 Problems 39 41 Find the equation of a line given the slope and a point contained in the line Sec 3 4 Obj 1 p 156 To determine the equation of a straight line given a set of condi tions we c
40. e given point x y1 we can determine slope to be m 2 n where x x i Simplifying gives us the equation y y m x x Figure 3 42 We refer to the equation y y m x x as the point slope form of the equation of a straight line Instead of the approach we used in Example 1 we could use the point slope form to write the equation of a line with a given slope that contains a given point 4 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 159 EXAMPLE 4 o REVISED PAGES 3 4 Determining the Equation of a Line 159 3 Use the point slope form to find the equation of a line that has a slope of 5 and con tains the point 2 4 Solution 3 We can determine the equation of the line by substituting 5 for m and 2 4 for x y1 in the point slope form y Y mx x 3 y 4 x 2 S y 4 3 x 2 Sy 20 3x 6 14 3x 5y Thus the equation of the line is 3x 5y 14 Y PRACTICE YOUR SKILL 4 Use the point slope form to find the equation of a line that has a slope of 3 and contains the point 2 5 4x 3y 23 E Apply the Slope Intercept Form of an Equation Another special form of the equation of a line is the slope intercept form Let s use the point slope form to find the equation of a line that has a slope of m and a y intercept of b A y intercept of b means that the line contains the point 0 b as in Figure 3 43 Therefore we can use the p
41. e horizontal axis c label the vertical axis s and use the origin along with one ordered pair from the table to pro duce the straight line graph in Figure 3 11 Because of the type of application we use only nonnegative values for c and s Figure 3 11 From the graph we can approximate s values on the basis of given c values For example if c 30 then by reading up from 30 on the c axis to the line and then across to the s axis we see that s is a little less than 40 An exact s value of 39 is obtained by using the equation s 1 3c Many formulas that are used in various applications are linear equations in 3 two variables For example the formula C gf 32 which is used to convert temperatures from the Fahrenheit scale to the Celsius scale is a linear relation ship Using this equation we can determine that 14 F is equivalent to 5 5 5 C 9 14 32 9 18 10 C Let s use the equation C gE 32 to com plete the following table F 22 13 3 32 50 68 86 C 30 25 15 0 10 20 30 Reading from the table we see for example that 13 F 25 C and 68 F 20 C 5 To graph the equation C oF 32 we can label the horizontal axis F label the vertical axis C and plot two ordered pairs F C from the table Fig ure 3 12 shows the graph of the equation From the graph we can approximate C values on the basis of given F values For example if F 80 then by reading u
42. e line through 1 y and 4 2 has a slope 5 f 3 3 Find x if the line through x 4 and 2 5 has a slope of 2 4 Find y if the line through 5 2 and 3 y has a slope 7 p ig Tg For Problems 33 40 you are given one point on a line and the slope of the line Find the coordinates of three other points on the line Answers vary 33 2 5 P 34 aam i gt gt e i i 6 35 3 4 m 3 36 3 6 m 1 37 39 2 3 6 2 m 3 38 4 1 m 7 2 4 m 2 40 5 3 m 3 b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 153 For Problems 41 50 find the coordinates of two points on the given line and then use those coordinates to find the slope of the line P41 2x 3y 6 42 4x 5y 20 2 43 x 2y 4 3 gt 44 3x y 12 3 45 4x 7y 12 46 2x 7y 11 a gt 47 y 4 0 48 x 3 Undefined gt 49 y 5x 5 50 y 6x 0 6 Use Slope to Graph Lines For Problems 51 58 graph the line that passes through the given point and has the given slope See answer section 3 52 1 0 m 51 3 1 m 3 4 53 2 3 m 1 54 1 4 m 3 55 0 5 56 3 4 A m 3 m 4 2 57 2 2 58 3 4 a _ m m i 2 2 p REVISED PAGES 3 3 Distance and Slope 153 Apply Slope to Solve Problems gt 59 gt 60 Pol gt 62 gt 64 A certain highway has a 2 grade How many feet does it rise in a h
43. e points a 3 1 and 7 5 5 3 b 2 8 and 6 4 2 6 c 3 2 and 5 8 1 5 d 4 10 and 9 25 z 2 e 4 1 and 10 5 7 2 5 8 and 1 7 2 8 GRAPHING CALCULATOR ACTIVITIES 71 Remember that we did some work with parallel lines back in the graphing calculator activities in Problem Set 3 1 Now let s do some work with perpendicular lines Be sure to set your boundaries so that the distance between tick marks is the same on both axes 1 a Graph y 4x and y 7 on the same set of axes Do they appear to be perpendicular lines 1 b Graph y 3x and y 3 on the same set of axes Do they appear to be perpendicular lines 2 5 c Graph y 57 1 and y 7 2 on the same set of axes Do they appear to be perpendi cular lines 18 47 9 15 69 d 1 2 e Z 7 f g gt 72 3 4 4 d Graph y 7 3 y 3 2 and y 3x 2 on the same set of axes Does there appear to be a pair of perpendicular lines e On the basis of your results in parts a through d make a statement about how we can recognize per pendicular lines from their equations For each of the following pairs of equations 1 predict whether they represent parallel lines perpendicular lines or lines that intersect but are not perpendicular and 2 graph each pair of lines to check your prediction a 5 2x 3 3y 9 4 and 5 2x 3 3y 12 6 b 1 3x
44. e with vertices at 1 2 3 4 and 3 6 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 143 F REVISED PAGES 3 3 Distance and Slope 143 1 False 2 True 3 False 4 True 5 False 6 False 7 True 8 True 9 False 10 False 3 3 Distance and Slope OBJECTIVES Find the Distance between Two Points Find the Slope of a Line Use Slope to Graph Lines Apply Slope to Solve Problems Find the Distance between Two Points As we work with the rectangular coordinate system it is sometimes necessary to express the length of certain line segments In other words we need to be able to find the distance between two points Let s first consider two specific examples and then develop the general distance formula 4 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 144 F REVISED PAGES 144 Chapter 3 Linear Equations and Inequalities in Two Variables EXAMPLE 1 Find the distance between the points A 2 2 and B 5 2 and also between the points C 2 5 and D 2 4 Solution Let s plot the points and draw AB as in Figure 3 22 Because AB is parallel to the x axis its length can be expressed as 5 2 or 2 5 The absolute value is used to ensure a nonnegative value Thus the length of AB is 3 units Likewise the length of CD is 5 4 4 5 9 units Gap D 2 4 Figure 3 22 Y PRACTICE YOUR SKILL Find the distance between the points A 3 6 and B 3 2 8 units E EXAMPLE
45. hapter 3 Linear Equations and Inequalities in Two Variables Figure 3 19 Y PRACTICE YOUR SKILL Graph 3x y lt 3 See answer section EXAMPLE 2 Graph 3x 2y lt 6 Solution Step 1 Graph 3x 2y 6 as a solid line because equality is included in 3x 2y lt 6 Figure 3 20 Step 2 Choose the origin as a test point and substitute its coordinates into the given statement 3x 2y 6 becomes 3 0 2 0 6 which is true Step 3 Because the test point satisfies the given statement all points in the same half plane as the test point satisfy the statement Thus the graph of 3x 2y lt 6 consists of the line and the half plane below the line Figure 3 20 Figure 3 20 Y PRACTICE YOUR SKILL Graph x 4y 4 See answer section E 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 141 o REVISED PAGES 3 2 Linear Inequalities in Two Variables 141 EXAMPLE 3 Graph p Sy Solution Step 1 Graph y 3x as a solid line because equality is included in the statement y 3x Figure 3 21 Step 2 The origin is on the line so we must choose some other point as a test point Let s try 2 1 ys3x becomes 1 3 2 which is a true statement Step 3 Because the test point satisfies the given inequality the graph is the half plane that contains the test point Thus the graph of y 3x consists of the line and the half plane below the line as indicated in Figure 3 21 Figure 3 21 Y PRACTICE YOUR SKILL
46. he line that is perpendicular to the line x 6y 9 and con tains the point 4 7 Express the equation in standard form 6x y 31 The monthly bill for a cellular phone can be described by a linear relationship Find the equation for this relationship if the bill for 750 minutes used is 35 00 and the bill for 550 minutes used is 31 00 Let y represent the amount of the bill and let x represent the number of minutes used Write the equation in slope intercept form y gg 20 or y 0 02x 20
47. ing calculator or a computer The examples were chosen to reinforce concepts under discussion Courtesy Texas Instruments Figure 3 13 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 133 F REVISED PAGES 3 1 Rectangular Coordinate System and Linear Equations 133 EXAMPLE 8 Use a graphing utility to obtain a graph of the line 2 1x 5 3y 7 9 Solution First let s solve the equation for y in terms of x 2 1 5 3y 7 9 5 3y 7 9 2 1x 1 gt 21 53 i 7 9 21x Now we can enter the expression for Y and obtain the graph as ene 5 3 shown in Figure 3 14 10 15 15 10 Figure 3 14 V PRACTICE YOUR SKILL Use a graphing utility to obtain a graph of the line 3 4x 2 5y 6 8 E See answer section CONCEPT QUIZ For Problems 1 10 answer true or false 1 In a rectangular coordinate system the coordinate axes partition the plane into four parts called quadrants 2 Quadrants are named with Roman numerals and are numbered clockwise 3 The real numbers in an ordered pair are referred to as the coordinates of the point 4 Ifthe abscissa of an ordered pair is negative then the point is in either the 3rd or 4th quadrant 5 The equation y x 3 has an infinite number of ordered pairs that satisfy the equation 6 The graph of y x is a straight line 7 The y intercept of the graph of 3x 4y 4 is 4 8 The graph of y 4 is a vertical line 9 The graph of x 4 has an x interce
48. line through 4 3 and 12 y has a slope of 2 5 8 Find x if the line through x 5 and 3 1 has a slope 3 f 1 wa b p REVISED PAGES Chapter 3 Review Problem Set 175 For Problems 33 36 graph the line that has the indicated slope and contains the indicated point 33 35 37 38 39 40 41 See answer section 1 3 m gt 0 3 34 m 5 0 4 m 3 1 2 36 m 2 1 4 A certain highway has a 6 grade How many feet does it rise in a horizontal distance of 1 mile 5280 feet 316 8 ft gt If the ratio of rise to run is to be 3 for the steps of a stair case and the run is 12 inches find the rise 8 in Find the slope of each of the following lines 5 a 4x y 7 m 4 b 2x 7y 3 meg Find the slope of any line that is perpendicular to the line e y T gt 3 Find the slope of any line that is parallel to the line 4 4x 5y 10 a For Problems 42 49 write the equation of the line that satis fies the stated conditions Express final equations in standard form 42 43 44 45 46 47 49 50 51 Having a slope of 3 and a y intercept of 4 3x 7y 28 2 Containing the point 1 6 and having a slope of 2x 3y 16 3 Containing the point 3 5 and having a slope of 1 xty 2 Containing the points 1 2 and 3 5 7x 4y 1 Containing the points 0 4 and 2 6 x y 4 Containing the point
49. m 68 the formula for the o S A D C x coordinate of the midpoint is x x 3 x Figure 3 37 This formula can be manipulated algebraically to produce a simpler formula Therefore 1 x X 5 x 1 2 7 Hee es x T 3 3 1 1 aN AS A Re Similarly CB can be treated as a segment of a number line as shown in Figure 3 38 Therefore 1 1 Pelee oe eae Bes 2 2 2 2 y 2 7 5 2 2 3 4 Eey 3 3 Xx xy The coordinates of point P are 5 4 Ce2 Figure 3 38 For each of the following find the coordinates of the indi cated point in the xy plane a One third of the distance from 2 3 to 5 9 3 5 b Two thirds of the distance from 1 4 to 7 13 5 10 c Two fifths of the distance from 2 1 to 8 11 2 5 d Three fifths of the distance from 2 3 to 3 8 See e Five eighths of the distance from 1 2 tobelow 4 10 See below Seven eighths of the distance from 2 3 to 1 9 See below Suppose we want to find the coordinates of the midpoint of a line segment Let P x y represent the midpoint of the line segment from A x y to B x y2 Using Hence the x coordinate of the midpoint can be interpreted as the average of the x coordinates of the endpoints of the line segment A similar argument for the y coordinate of the midpoint gives the following formula th 2 For each of the pairs of points use the formula to find the midpoint of the line segment between th
50. mining the Equation of a Line 161 2 EXAMPLE 7 Graph the line determined by the equation y 37 1 Solution Comparing the given equation to the general slope intercept form we see that the 2 slope of the line is 3 and the y intercept is 1 Because the y intercept is 1 we can 2 plot the point 0 1 Then because the slope is 3 let s move 3 units to the right and 2 units up from 0 1 to locate the point 3 1 The two points 0 1 and 3 1 determine the line in Figure 3 44 Again you should determine a third point as a check point Figure 3 44 Y PRACTICE YOUR SKILL 1 Graph the line determined by the equation y at 2 See answer section E We use two forms of equations of straight lines extensively They are the standard form and the slope intercept form and we describe them as follows Standard Form Ax By C where B and C are integers and A is a nonneg ative integer A and B not both zero Slope Intercept Form y mx b where m is a real number representing the slope and b is a real number representing the y intercept 4 Find the Equations for Parallel or Perpendicular Lines We can use two important relationships between lines and their slopes to solve cer tain kinds of problems It can be shown that nonvertical parallel lines have the same slope and that two nonvertical lines are perpendicular if the product of their 03 W4928 AM1 qxd 11
51. n E The three parts of Example 5 represent the three basic possibilities for slope that is the slope of a line can be positive negative or zero A line that has a positive slope rises as we move from left to right as in Figure 3 28 A line that has a negative slope falls as we move from left to right as in Figure 3 29 A horizontal line as in Figure 3 30 has a slope of zero Finally we need to realize that the concept of slope is undefined for vertical lines This is due to the fact that for any vertical line the horizontal change as we move from one point on the line to another is zero Thus the ratio will have a denominator of zero and be undefined Accord 2 1 ingly the restriction x x is imposed in Definition 3 1 One final idea pertaining to the concept of slope needs to be emphasized The slope of a line is a ratio the ratio of vertical change to horizontal change 2 A slope of means that for every 2 units of vertical change there must be a corresponding 3 units of horizontal change Thus starting at some point on a line that 2 has a slope of 3 we could locate other points on the line as follows 2 4 a by moving 4 units up and 6 units to the right 2 8 yo by moving 8 units up and 12 units to the right 2 3573 gt by moving 2 units down and 3 units to the left b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 149 F REVISED PAGES 3 3 Distance and Slope 149 3 Likewise if a line has a
52. oblems are specifically designed for those of you who have access to a graphing calculator or a computer with an appropriate soft ware package Within the framework of these problems you will be given the opportunity to reinforce concepts we dis cussed in the text lay groundwork for concepts we will intro duce later in the text predict shapes and locations of graphs on the basis of your previous graphing experiences solve prob lems that are unreasonable or perhaps impossible to solve without a graphing utility and in general become familiar with the capabilities and limitations of your graphing utility 56 a Graph y 3x 4 y 2x 4 y 4x 4 and y 2x 4 on the same set of axes 1 b Graph y hal 3 y 5x 3 y 0 1x 3 and y 7x 3 on the same set of axes c What characteristic do all lines of the form y ax 2 where a is any real number share b gt 52 x y 1 53 E GRAPHING CALCULATOR ACTIVITIES x y 4 54 2x y 4 55 3x 2y 6 57 a Graph y 2x 3 y 2x 3 y 2x y 2x 5 on the same set of axes b Graph y 3x 1 y 3x 4 y 3x 2 and 6 and y 3x 5 on the same set of axes 1 1 c Graph y 5x H 3 y 3x 4y 5 5 and 1 yr a 2 on the same set of axes d What relationship exists among all lines of the form y 3x b where b is any real number 58 a Gr
53. ocated somewhere be tween two given points For example suppose that we want to find the coordinate x of the point located two thirds of the distance from 2 to 8 Because the total dis tance from 2 to 8 is 8 2 6 units we can start at 2 and 2 2 move 3 6 4 units toward 8 Thus x 2 3 9 2 4 6 For each of the following find the coordinate of the indi cated point on a number line a Two thirds of the distance from 1 to 10 7 b Three fourths of the distance from 2to14 10 c One third of the distance from 3 to 7 2 d Two fifths of the distance from 5 to 6 a e Three fifths of the distance from 1 to 11 7 Five sixths of the distance from 3 to 7 69 Now suppose that we want to find the coordinates of point P which is located two thirds of the distance from A 1 2 to B 7 5 in a coordinate plane We have plotted oe the given points A and B in Figure 3 36 to help with the analysis of this problem Point D is two thirds of the dis tance from A to C because parallel lines cut off propor tional segments on every transversal that intersects the lines Thus AC can be treated as a segment of a number line as shown in Figure 3 37 B 7 5 ae ana D x 2 AD CORY Figure 3 36 03 W4928 AM1 qxd 154 70 11 3 08 8 26 PM Page 154 p REVISED PAGES Chapter 3 Linear Equations and Inequalities in Two Variables 1 x 7 the method in Proble
54. oint slope form as follows y Y m x x y b m x 0 y b mx y mx b y 0 b x Figure 3 43 oe 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 160 F REVISED PAGES 160 Chapter 3 Linear Equations and Inequalities in Two Variables We refer to the equation y mx b as the slope intercept form of the equation of a straight line We use it for three pri mary purposes as the next three examples illustrate 1 EXAMPLE 5 Find the equation of the line that has a slope of 4 and a y intercept of 2 Solution This is a restatement of Example 3 but this time we will use the slope intercept _ 1 form y mx b of a line to write its equation Because m 4 and b 2 we can substitute these values into y mx b y mx b 1 yugxt2 4y x 8 Multiply both sides by 4 x 4y 8 Same result as in Example 3 Y PRACTICE YOUR SKILL Find the equation of the line that has a slope of 3 and a y intercept of 8 E 3x y 8 EXAMPLE 6 Find the slope of the line when the equation is 3x 2y 6 Solution We can solve the equation for y in terms of x and then compare it to the slope intercept form to determine its slope Thus 3x 2y 6 2y 3x 6 3 y rta 3 yoy 8 y mx b Pees ee 3 The slope of the line is gt Furthermore the y intercept is 3 Y PRACTICE YOUR SKILL Find the slope of the line when the equation is 4x 5y 10 m f E b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 161 REVISED PAGES 3 4 Deter
55. or Problems 23 28 the situations can be described by the use of linear equations in two variables If two pairs of values are known then we can determine the equation by using the approach used in Example 2 of this section For each of the fol lowing assume that the relationship can be expressed as a lin ear equation in two variables and use the given information to determine the equation Express the equation in slope intercept form 23 A company uses 7 pounds of fertilizer for a lawn that measures 5000 square feet and 12 pounds for a lawn that gt Blue arrows indicate Enhanced WebAssi 3 3x y 3 gn problems gt 24 25 26 27 28 measures 10 000 square feet Let y represent the pounds of fertilizer and x the square footage of the lawn y x2 4000 A new diet fad claims that a person weighing 140 pounds should consume 1490 daily calories and that a 200 pound person should consume 1700 calories Let y represent the calories and x the weight of the person in pounds y Zx 1000 Two banks on opposite corners of a town square had signs that displayed the current temperature One bank dis played the temperature in degrees Celsius and the other in degrees Fahrenheit A temperature of 10 C was dis played at the same time as a temperature of 50 F On an other day a temperature of 5 C was displayed at the same time as a temperature of 23 F Let y represent the temperature in degrees Fahrenheit and
56. ordinate system is the polar coordinate system Conduct an Internet search to see an example of the polar coordinate system How are the coordinates of a point determined in the polar coordinate 4 system Rectangular Coordinate System and Linear Equations OBJECTIVES Find Solutions for Linear Equations in Two Variables Review of the Rectangular Coordinate System Graph the Solutions for Linear Equations Graph Linear Equations by Finding the x and y Intercepts Graph Lines Passing through the Origin Vertical Lines and Horizontal Lines Apply Graphing to Linear Relationships G 0o nN amp Introduce Graphing Utilities Optional Exercises El Find Solutions for Linear Equations in Two Variables In this chapter we want to consider solving equations in two variables Let s begin by considering the solutions for the equation y 3x 2 A solution of an equation in two variables is an ordered pair of real numbers that satisfies the equation When using the variables x and y we agree that the first number of an ordered pair is a value of x and the second number is a value of y We see that 1 5 is a solution for y 3x 2 because if x is replaced by 1 and y by 5 the result is the true numerical statement 5 3 1 2 Likewise 2 8 is a solution because 8 3 2 2 is a true numerical statement We can find infinitely many pairs of real numbers that satisfy y 3x 2 by arbitrarily choosing values for x and then for ea
57. orizontal distance of 1 mile 1 mile 5280 feet 105 6 ft The grade of a highway up a hill is 30 How much change in horizontal distance is there if the vertical height of the hill is 75 feet 250 ft Suppose that a highway rises a distance of 215 feet in a horizontal distance of 2640 feet Express the grade of the highway to the nearest tenth ofa percent 8 1 3 If the ratio of rise to run is to be for some steps and the rise is 19 centimeters find the run to the nearest centimeter 32cm 2 If the ratio of rise to run is to be for some steps and the run is 28 centimeters find the rise to the nearest centimeter 19cm 1 Suppose that a county ordinance requires a 27 fall for a sewage pipe from the house to the main pipe at the street How much vertical drop must there be for a hori zontal distance of 45 feet Express the answer to the near est tenth of a foot 1 0 ft A THOUGHTS INTO WORDS 65 How would you explain the concept of slope to someone who was absent from class the day it was discussed 2 gt 66 If one line has a slope of 5 and another line has a slope 3 of 7 which line is steeper Explain your answer 67 2 Suppose that a line has a slope of 3 and contains the point 4 7 Are the points 7 9 and 1 3 also on the line Explain your answer FURTHER INVESTIGATIONS gt 68 Sometimes it is necessary to find the coordinate of a point on a number line that is l
58. p from 80 on the F axis to the line and then across to the C axis we see that C is approximately 25 Likewise we can obtain ap proximate F values on the basis of given C values For example if C 25 then by b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 132 p REVISED PAGES 132 Chapter 3 Linear Equations and Inequalities in Two Variables reading across from 25 on the C axis to the line and then up to the F axis we see that F is approximately 15 Figure 3 12 Introduce Graphing Utilities The term graphing utility is used in current literature to refer to either a graphing cal culator see Figure 3 13 or a computer with a graphing software package We will frequently use the phrase use a graphing calculator to mean use a graphing calcula tor or a computer with the appropriate software These devices have a large range of capabilities that enable the user not only to obtain a quick sketch of a graph but also to study various characteristics of it such as the x intercepts y intercepts and turning points of a curve We will introduce some of these features of graphing utilities as we need them in the text Because there are so many different types of graphing utilities available we will use mostly generic termi nology and let you consult your user s manual for specific key punching instructions We urge you to study the graphing utility examples in this text even if you do not have access to a graph
59. plete because the pitch is so great that scaffolding will be required Figure 3 33 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 151 F REVISED PAGES 3 3 Distance and Slope 151 The concept of slope is also used in the construction of flights of stairs Fig ure 3 34 The terms rise and run are commonly used and the steepness slope of the stairs can be expressed as the ratio of rise to run In Figure 3 34 the stairs on the 10 left where the ratio of rise to run is Th are steeper than the stairs on the right which 7 h tio of ave a ratio of 57 Figure 3 34 In highway construction the word grade is used for the concept of slope For example in Figure 3 35 the highway is said to have a grade of 17 This means that for every horizontal distance of 100 feet the highway rises or drops 17 feet In other 17 words the slope of the highway is 700 17 feet 100 feet Figure 3 35 EXAMPLE 8 A certain highway has a3 grade How many feet does it rise in a horizontal distance of 1 mile Solution A 3 grade means a slope of ae Therefore if we let y represent the unknown vertical distance and use the fact that 1 mile 5280 feet we can set up and solve the following proportion CUREA 100 5280 100y 3 5280 15 840 y 158 4 The highway rises 158 4 feet in a horizontal distance of 1 mile Y PRACTICE YOUR SKILL A certain highway has a 2 5 grade How many feet doe
60. pt of 4 10 The graph of every linear equation has a y intercept 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 134 134 p REVISED PAGES Chapter 3 Linear Equations and Inequalities in Two Variables Problem Set 3 1 El Find Solutions for Linear Equations in Two Variables For Problems 1 4 determine which of the ordered pairs are solutions to the given equation 1 y 3x 2 2 4 1 5 0 1 2 4 ER 5 2 y 2x 3 2 5 1 5 1 1 1 5 1 1 3 2x y 6 2 10 1 5 3 0 11 1 4 3x 2y 2 3 U2 2 1 a ak 3 Ne 2 2 1 x 2 10 3 0 Graph Lines Passing through the Origin Vertical Lines and Horizontal Lines For Problems 29 40 graph each of the linear equations See answer section gt 29 gt 31 gt 33 P35 gt 37 gt 39 y y 3x 2x 3y x 0 y 2 x 4 30 32 0 34 36 38 40 y x y 4x 3x 4y 0 y 0 x 3 y l1 Graph the Solutions for Linear Equations A Apply Graphing to Linear Relationships P41 a Digital Solutions charges for help desk services according to the equation c 0 25m 10 where c represents the cost in dollars and m represents the For Problems 5 8 complete the table of values for the equa tion and graph the equation See answer section x 1 0 4 minutes of service Complete the following table 5 y x 3 y 5 a m 5 10 15 20 30 60 2 0 2 c 11 25 12 50 13 75 15 00 17 50
61. r Problems 67 78 write the equation of the line that sat isfies the given conditions Express final equations in stan dard form gt 67 Contains the point 2 4 and is parallel to the y axis x 0y 2 68 Contains the point 3 7 and is parallel to the x axis KF Y 7 gt 69 Contains the point 5 6 and is perpendicular to the y axis Ox y 6 70 Contains the point 4 7 and is perpendicular to the xaxis x 0y 4 71 Contains the point 1 3 and is parallel to the line x 5y 9 x 5y 16 gt 72 Contains the point 1 4 and is parallel to the line x 2y 6 x 2y 9 73 Contains the origin and is parallel to the line 4x 7y 3 4x 7y 0 74 Contains the origin and is parallel to the line 2x 9y 4 2x 9y 0 gt 75 Contains the point 1 3 and is perpendicular to the line 2x y 4 x 2y 5 76 Contains the point 2 3 and is perpendicular to the linex 4y 6 4x y 5 gt 77 Contains the origin and is perpendicular to the line 2x 3y 8 3x 2y 0 78 Contains the origin and is perpendicular to the line y 5x x 5y 0 THOUGHTS INTO WORDS 79 What does it mean to say that two points determine a line 80 How would you help a friend determine the equation of the line that is perpendicular to x Sy 7 and contains the point 5 4 3 9 1 w lt N 81 Explain how you would find the slope of the line y 4 45 m b 46 m zb 47 m b 48
62. r would be visible on the screen when we looked for certain values Now let s use the TRACE feature of the graphing cal culator to complete the following table Note that the cur sor moves in increments of 1 as we trace along the graph F 5 5 9 C 11 12 20 30 45 60 This was accomplished by setting the aforementioned fraction equal to 1 By moving the cursor to each of the F values we can complete the table as follows F 5 5 9 11 12 20 30 45 60 C 1 21 15 13 12 11 7 1 7 16 The C values are expressed to the nearest degree Use your calculator and check the values in the table by s 5 using the equation C oF 32 9 60 a Use your graphing calculator to graph F zC 32 Be sure to set boundaries on the horizontal axis so that when you are using the trace feature the cursor will move in increments of 1 b Use the TRACE feature and check your answers for part a of Problem 42 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 137 F REVISED PAGES 3 1 Rectangular Coordinate System and Linear Equations 137 bk 4 y EE 3 0 5 3 0 X 5 y 6 3 4 T y 3x 0 0 3 0 x GRD 7 y 8 y y 4 0 4 2 4 X X 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 138 F REVISED PAGES 138 Chapter 3 Linear Equations and Inequalities in Two Variables 3 2 Linear Inequalities in Two Variables OBJECTIVES El Graph Linear Inequalities El Graph Linear Inequalities Linear inequalities in two variables
63. rdered pair 3 5 means that the point D is located three units to the left and five units down from the origin b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 124 p REVISED PAGES 124 Chapter 3 Linear Equations and Inequalities in Two Variables DEMES e Figure 3 2 Remark The notation 2 4 was used earlier in this text to indicate an interval of the real number line Now we are using the same notation to indicate an ordered pair of real numbers This double meaning should not be confusing because the context of the material will always indicate which meaning of the notation is being used Throughout this chapter we will be using the ordered pair interpretation In general we refer to the real numbers a and b in an ordered pair a b associated with a point as the coordinates of the point The first number a called the abscissa is the directed distance of the point from the vertical axis measured parallel to the hori zontal axis The second number b called the ordinate is the directed distance of the point from the horizontal axis measured parallel to the vertical axis Figure 3 3a Thus in the first quadrant all points have a positive abscissa and a positive ordinate In the second quadrant all points have a negative abscissa and a positive ordinate We have indicated the sign situations for all four quadrants in Figure 3 3 b This system of as sociating points in a plane with pairs of real numbers is calle
64. relationships Sec 3 1 Obj 6 p 130 quantities are linear relationships Graphs of these relationships can be used to present information about the relationship dollars and let w represent the gallons of water used then the equation c 0 004w 20 can be used to determine the cost of a water bill for a household Graph the relationship Solution Label the vertical axis c and the horizontal axis w Because of the type of application we use only nonnegative values for w c 0 004w 20 2000 4000 w Graph linear inequalities Sec 3 2 Obj 1 p 138 To graph a linear inequality first graph the line for the corresponding equality Use a solid line if the equality is included in the given statement or a dashed line if the equality is not included Then a test point is used to determine which half plane is included in the solution set See page 139 for the detailed steps Graph x 2y lt 4 Solution First graph x 2y 4 Choose 0 0 as a test point Substituting 0 0 into the inequality yields 0 4 Because the test point 0 0 makes the inequality a false statement the half plane not containing the point 0 0 is in the solution Problems 21 26 continued 03 W4928 AM1 qxd 11 3 08 172 OBJECTIVE 8 27 PM Page 172 SUMMARY p REVISED PAGES Chapter 3 Linear Equations and Inequalities in Two Variables EXA
65. s it rise in a horizontal dis tance of 2000 feet 50 ft b 03 W4928 AM1 qxd 152 11 3 08 8 26 PM Page 152 p REVISED PAGES Chapter 3 Linear Equations and Inequalities in Two Variables CONCEPT Q UIZ For Problems 1 10 answer true or false 1 When applying the distance formula V x x1 y2 y to find the dis tance between two points you can designate either of the two points as P Pie E An isosceles triangle has two sides of the same length The distance between the points 1 4 and 1 2 is 2 units The distance between the points 3 4 and 3 2 is undefined The slope of a line is the ratio of the vertical change to the horizontal change when moving from one point on the line to another point on the line x The slope of a line is always positive 7 A slope of 0 means that there is no change in the vertical direction when mov ing from one point on the line to another point on the line 8 The concept of slope is undefined for horizontal lines 9 When applying the slope formula m 2 Mt to find the slope of a line X2 1 between two points you can designate either of the two points as P 10 then the run is 6 inches f 2 59 bh aah If the ratio of the rise to the run for some steps is 4 and the rise is 9 inches Problem Set 3 3 Find the Distance between Two Points For Problems 1 12 find the distance between each of the pairs of points Express
66. t is a line parallel to the x axis that has a y intercept of b d Apply Graphing to Linear Relationships There are numerous applications of linear relationships For example suppose that a retailer has a number of items that she wants to sell at a profit of 30 of the cost of each item If we let s represent the selling price and c the cost of each item then the equation s c 0 3c 1 3c can be used to determine the selling price of each item based on the cost of the item In other words if the cost of an item is 4 50 then it should be sold for s 1 3 4 5 5 85 The equation s 1 3c can be used to determine the following table of values Reading from the table we see that if the cost of an item is 15 then it should be sold for 19 50 in order to yield a profit of 30 of the cost Furthermore because this is a linear relationship we can obtain exact values between values given in the table c 1 5 10 15 20 s 13 6S 13 195 26 b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 131 p REVISED PAGES 3 1 Rectangular Coordinate System and Linear Equations 131 For example a c value of 12 5 is halfway between c values of 10 and 15 so the corresponding s value is halfway between the s values of 13 and 19 5 Therefore a c value of 12 5 produces an s value of s 13 5 19 13 16 25 Thus if the cost of an item is 12 50 it should be sold for 16 25 Now let s graph this linear relationship We can label th
67. ter Now let s analyze some problems of type 2 that deal specifically with straight lines Given certain facts about a line we need to be able to determine its algebraic equation Let s consider some examples 2 EXAMPLE 1 Find the equation of the line that has a slope of 3 and contains the point 1 2 Solution First lets draw the line and record the given information Then choose a point x y that represents any point on the line other than the given point 1 2 See Fig ure 3 39 The slope determined by 1 2 and x y is Thus y 2_2 x 1 3 x 1 3 y 2 2x 2 3y 6 2x 3y 4 Figure 3 39 Y PRACTICE YOUR SKILL 3 Find the equation of the line that has a slope of 4 and contains the point 3 1 E 3x 4y 13 b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 157 F REVISED PAGES 3 4 Determining the Equation of a Line 157 Find the Equation of a Line Given Two Points EXAMPLE 2 Find the equation of the line that contains 3 2 and 2 5 Solution First let s draw the line determined by the given points Figure 3 40 if we know two points we can find the slope m 22 3 3 X2 Xi 5 5 Now we can use the same approach as in Example 1 Form an equation using a variable point x y one of the two given points and the slope of Tz y 5_3 2 x 2 5 5 5 3 x 2 S5 y 5 Figure 3 40 3x 6 5y 25 3x 5y 19 WY PRACTICE
68. the slope of 2x y 6 Let s find the slope of 2x y 6 by changing it to the slope intercept form b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 164 F REVISED PAGES 164 Chapter 3 Linear Equations and Inequalities in Two Variables 2x y 6 y 2x 6 y 2x 6 The slope is 2 Figure 3 46 1 The slope of the desired line is 7 the negative reciprocal of 2 and we can proceed as before by using a variable point x y y 2 1 xt 1 2 I x 1 2 y 2 x 1 2y 4 x 2y 5 Y PRACTICE YOUR SKILL Find the equation of the line that contains the point 3 1 and is perpendicular to the line determined by 5x 2y 10 2x 5y 11 CONCEPT Q UIZ For Problems 1 10 answer true or false If two lines have the same slope then the lines are parallel If the slopes of two lines are reciprocals then the lines are perpendicular In the standard form of the equation of a line Ax By C A can be a rational number in fractional form In the slope intercept form of an equation of a line y mx b mis the slope In the standard form of the equation of a line Ax By C A is the slope The slope of the line determined by the equation 3x 2y 4 is 7 The concept of a slope is not defined for the line y 2 The concept of slope is not defined for the line x 2 The lines determined by the equations x 3y 4 and 2x 6y 11 are parallel lines The lines determine
69. tical axis c graph the equation c 75h 150 for nonnegative values of h See answer section c Use the graph from part b to approximate values of c when h 1 5 and 3 5 d Check the accuracy of your reading from the graph in part c by using the equation c 75h 150 a The value added tax is computed by the equation t 0 15v where t represents the tax and v represents the value of the goods Complete the following table v 100 200 350 400 t 15 30 52 50 60 b Labeling the horizontal axis v and the vertical axis t graph the equation 0 15v for nonnegative values of y See answer section c Use the graph from part b to approximate values of t when v 250 and v 300 d Check the accuracy of your reading from the graph in part c by using the equation 0 15v For Problems 21 26 graph each inequality See answer section 21 23 25 27 28 29 30 31 32 x 3y lt 6 22 x 2y 4 1 2x 3y 6 24 yo ax 3 2 y lt 2x 5 26 ae Find the distance between each of the pairs of points a 1 5 and 1 2 V53 b 5 0 and 2 7 58 Find the lengths of the sides of a triangle whose vertices are at 2 3 5 1 and 4 5 5 10 V97 Verify that 1 2 is the midpoint of the line segment join ing 3 1 and 5 5 Find the slope of the line determined by each pair of points 3 4 2 2 E 2 3 4 1 6 5 Find y if the
70. wo slopes is 1 is the same as saying 1 that the two slopes are negative reciprocals of each other that is m m b 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 163 F REVISED PAGES 3 4 Determining the Equation of a Line 163 EXAMPLE 9 Find the equation of the line that contains the point 1 4 and is parallel to the line determined by x 2y 5 Solution First let s draw a figure to help in our analysis of the problem Figure 3 45 Because the line through 1 4 is to be parallel to the line determined by x 2y 5 it must have the same slope Let s find the slope by changing x 2y 5 to the slope intercept form x 2y 5 2y x 5 1 5 a aa Figure 3 45 1 The slope of both lines is ma Now we can choose a variable point x y on the line through 1 4 and proceed as we did in earlier examples y 4 1 x 1 2 I x 1 2 y 4 x 1 2y 8 x 2y 9 Y PRACTICE YOUR SKILL Find the equation of the line that contains the point 2 7 and is parallel to the line determined by 3x y 4 3x y 1 E EXAMPLE 10 Find the equation of the line that contains the point 1 2 and is perpendicular to the line determined by 2x y 6 Solution First let s draw a figure to help in our analysis of the problem Figure 3 46 Because the line through 1 2 is to be perpendicular to the line determined by 2x y 6 its slope must be the negative reciprocal of
71. x y 1 and thus is linear and produces a straight line graph The knowledge that any equation of the form Ax By C produces a straight line graph along with the fact that two points determine a straight line makes graphing linear equations a simple process We merely find two solutions such as the intercepts plot the corresponding points and connect the points with a straight line It is usually wise to find a third point as a check point Let s consider an example EXAMPLE 3 Graph 3x 2y 12 Solution First let s find the intercepts Let x 0 then 3 0 2y 12 2y 12 y 6 Thus 0 6 is a solution Let y 0 then 3x 2 0 12 3x 12 x 4 Thus 4 0 is a solution Now let s find a third point to serve as a check point Let x 2 then 3 2 2y 12 6 2y 12 2y 6 y gt 3 Thus 2 3 is a solution Plot the points associated with these three solutions and connect them with a straight line to produce the graph of 3x 2y 12 in Figure 3 6 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 128 F REVISED PAGES 128 Chapter 3 Linear Equations and Inequalities in Two Variables x intercept Ri Check point 0 6 N y intercept Figure 3 6 Y PRACTICE YOUR SKILL Graph 3x y 6 See answer section Let s review our approach to Example 3 Note that we did not solve the equa tion for y in terms of x or for x in terms of y Because we know the graph is a
72. x the temperature in degrees Celsius y ax 32 An accountant has a schedule of depreciation for some business equipment The schedule shows that after 12 months the equipment is worth 7600 and that after 20 months it is worth 6000 Let y represent the worth and x represent the time in months y 200x 10 000 A diabetic patient was told on a doctor visit that her HA1c reading of 6 5 corresponds to an average blood glu cose level of 135 At the next checkup three months later the patient had an HA1c reading of 6 0 and was told that it corresponds to an average blood glucose level of 120 Let y represent the HA1c reading and let x represent the average blood glucose level y 30x 60 Hal purchased a 500 minute calling card for 17 50 After he used all the minutes on that card he purchased an other card from the same company at a price of 26 25 for 750 minutes Let y represent the cost of the card in dollars and let x represent the number of minutes y 0 035x Find the Equation of a Line Given the Slope and y Intercept For Problems 29 36 write the equation of the line that has the indicated slope m and y intercept b Express final equa tions in slope intercept form gt 29 31 33 35 3 2 m b 4 30 m b 6 See below See below m 2 b 3 32 m 3 b 1 y 2x 3 y 3x 1 2 3 M b 1 34 m 7 b 4 See below See below 1 oe See vic See m 0 b 4 Pow 36 m 5 a Boni 29 y 3x 4 30 y 2x 6
73. y intercept of the graph Let s define in general the intercepts of a graph 03 W4928 AM1 qxd 11 3 08 8 26 PM Page 127 F REVISED PAGES 3 1 Rectangular Coordinate System and Linear Equations 127 It is advantageous to be able to recognize the kind of graph that a certain type of equation produces For example if we recognize that the graph of 3x 2y 12isa straight line then it becomes a simple matter to find two points and sketch the line Let s pursue the graphing of straight lines in a little more detail In general any equation of the form Ax By C where A B and C are constants A and B not both zero and x and y are variables is a linear equation and its graph is a straight line Two points of clarification about this description of a linear equation should be made First the choice of x and y for variables is arbi trary Any two letters could be used to represent the variables For example an equa tion such as 3r 2s 9 can be considered a linear equation in two variables So that we are not constantly changing the labeling of the coordinate axes when graphing equations however it is much easier to use the same two variables in all equations Thus we will go along with convention and use x and y as variables Second the phrase any equation of the form Ax By C technically means any equation of the form Ax By C or equivalent to that form For example the equation y 2x 1 is equivalent to 2

3 Linear Equations and Inequalities in Two Variables

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