1.4 'til,, in TwoVariables LinearEquations 39 I I f) Replace W and A with your actual weight and height. Now draw a graph showing how B dependson height for suitablevalues of H. s) For a fixed weight and age,doesthe basic energyrequirementincreaseor decrease for taller persons?Increases h) For the three graphs that you have drawn, explain how to determine whether the graph is increasingor decreasingfrom the formula for the graph. \ *ir I In Section 1.3 we graphedlines, including horizontaland vertical lines.We learned that every line has an equation in standardform Ax * By : C. In this section we will continueto study lines. 1x2, !t) Slope of a line A road that has a 5o/ograde rises 5 feet for every horizontal run of 100 feet. A roof that has a 5-12 pitch rises 5 feet for every horizontal run of 12 feet.The grade of a The steepnessor slope of road and the pitch of a roof are measurementsof steepness. a line in the xy-coordinatesystemis the ratio of the rise (the changein y-coordinates) to the run (the changein x-coordinates)betweentwo points on the line: _ slooe ' Figure 1.27 Definition:Slope changeinl-coordinates_ rise changein r-coordinates mn If (x1,y1) and (*2,y) are the coordinatesof the two points in Fig. l.27,then the rise rs lz - y1 andthe run is x2 - x1: Theslopeofthe1inethrough(xr,,1J'1"z,yz)withx1*x2is lz-lr xz-xt i Note that if (x1, y) and (*t, yr) are two points for which xl : x2 then the line through them is a vertical line. Sincethis caseis not includedin the definition of slope,a vertical line doesnot havea slope.We also saythat the slopeof a vertical line is undefined. If we choosetwo points on a horizontal line then y1 : 12dnd lz - lt : 0. For any horizontalline the rise betweentwo points is 0 and the slopeis 0. 40 Chapter 1 ruffiffi Equations, Inequalities, and Modeling e**rr./t E Finding the stope In eachcasefind the slopeof the line that containsthe two givenpoints. a. (-3,4), (-1, -2) b . ( - 3 , 7 ) , ( 5 , 7 ) c . ( - 3 , 5 ) ,( - 3 , 8 ) $olution a. Use(*vyr) : (-3, 4) and(*z,yz) : (- l, -2) in theformula: -2-4 . rl-Vr s l o p e: ' r ; : lxq, y q) _l _ (_3) The slopeof the line is -3. b. Use (*v y) : ( -3, 7) and (rr, yr) : . vt-lr -6 - -3 2 (5, 7) in the formula: 7-7 slope:-;=:5_(_"r:8:o (xs,y) : 0 The slope of this horizontal line is 0. c. The line through (-3, 5) and (-3, 8) is a vertical line and so it does not have a slope. I (xvy) Figure1.28 The slope of a line is the same number regardless of which two points on the line are used in the calculation of the slope. To understandwhy, consider the two triangles shown in Fig. 1.28. These triangles have the same shapeand are called similar triangles.Becausethe ratios of correspondingsidesof similar trianglesare equal, the ratio of rise to run is the same for either triangle. Point-StopeForm Supposethata line through(xr, y1) hasslopem. Everyotherpoint (r, y) on the line mustsatisff the equation =:* becauseanytwo pointscanbe usedto find the slope.Multiply both sidesby x - x1 to get! - lr : m(x - xr),whichis thepoint-slopeform of the equationof a line. Theorem: Point-SlopeForm In Section 1.3 we started with the equation of a line and graphed the line. Using the point-slope form, we can start with a graph of a line or a description of the line and write the equation for the line. 1.4 wmtur LinearEquations in TwoVariables e-*r"^fb 41 A The equation of a tine given two points In each case graph the line through the given pair of points. Then find the equation of the line and solve itfor y if possible. c . ( 3 ,- 1 ) , ( 3 , 9 ) ^ . ( - 1 , 4 ) , ( 2 , 3 ) b . ( 2 , 5 ) ,( - 6 , 5 ) $olutisn 246810 a. Find the slope of the line shown in Fig. t.29 as follows: !z-!t "'-*r-x1 Figure1.29 3-4 -l 1 2-(-l) 3 3 Now use a point, say(2,3), and m - -{ !-lr:m(x-xr) - -; L y 3 (x - ;2 y - 3 : -r, + I -:7 4 -5 -4 -3 -2 * i* 2) in ttt. point-slope form: The equation in point-slope form i ll The equation solved fory T b. The slope of the line through (2,5) and (-6, 5) shown in Fig. 1.30 is 0. The equationof this horizontal line is y : 5. c. The line through (3, - 1) and (3, 9) shown in Fig. 1.31is vertical and it doesnot I have slope. Its equationisx : 3. Stope-InterceptForm Theline y : mx * bgoesthrough(0,b) and (7,m t b). Betweenthesetwopoints the rise is m andthe run is 1. So the slope is ru. Since (0, b) is the y-intercept and m is the slope, y : mx * b is called slope-intercept form. Any equation in standard form Ax * By : C canbe rewritten in slope-interceptform by solving the equation fory provided B + 0.lf B : 0, then the line is vertical and has no slope. F i g u r e1 . 3 1 I I i Theorem: Slope-lnterceptForm i*m;;ry.*ffffi;;ry'i Every nonvertical line has an equation in slope-intercept form. i:' !l r1..",,, - If you know the slope and y-intercept for a line, then you can use slope-intercept form to write its equation. For example, the equation of the line through (0, 9) with Chapter1 ffiwii# Equations, Inequalities, and Modeling slope4 is y - 4x * 9. In the next examplewe use the slope-interceptform to determine the slope andy-intercept for a line. E Find the stopeand y-intercept Ar*r""N, Identi$r the slope andy-intercept for the line 2x - 3y : 6. $otution First solve the equationfory to get it in slope-interceptform: 2x-3y:6 -31,y : -2x -t 6 2 !:1x-2 Figure1.32 So the slope is ] and they-intercept is (0, -2). I Using Stopeto Graph a line Slope is the ratio ffi that results from moving from one point to another on a line. A positive rise indicatesa motion upward and a negativerise indicatesa motion downward. A positive run indicates a motion to the right and a negative run indicates a motion to the left. If you start at anypoint on a line with slope thenmoving up ), 1 unit and2 units to the right will bring you back to the line. On a line with slope -3 F i g u r e1 . 3 3 ot t',moving down 3 units and 1 unit to the right will bring you back to the line. e**r"./t A Graphinga tine using its stope and y-intercept Graph eachline. a.y:3x- I b. 2 1x+4 $olution - t hasy-intercept (0, -1)andslope: or (0, -1) J. Startingat we obtain a secondpoint on the line by moving up 3 units and I unit to the right. So the line goesthrough (0, - 1) and (1,2) as shown inFig. 1.32. -!, * 4 hasy-i ntercept(0,4) and sl ope -?otf. . S tart ingat b . T h e \i n ey : (0, 4) we obtain a secondpoint on the line by moving down 2 units and then 3 units to the right. So the line goesthrough (0, 4) and (3, 2) as shownin Fig. 1.33. I a. The liney:3x Graphsof y=vv1;s , Figure1.34 As the .r-coordinateincreaseson a line with positive slope, the y-coordinate increasesalso. As the x-coordinate increaseson a line with negative slope, the y-coordinatedecreases.Figure 1.34 showssome lines of the form y : mx with positive slopesand negativeslopes.Observethe effect that the slope has on the position of the line. t; L- 1.4 ffiffii4rniLinearEquationsin Two Variables l, The Three Formsfor the Equationof a Line There are three forms for the equation of a line. The following strategywill help you decidewhen and how to use theseforms. STRATEGY Findingthe Equationof a Line 1. 2. 3. 4. 5. A-*rr"/t tr $tandardform using integers Find the equationof the line throurn (O,J) *i.n slope]. Wrlt. the equationin standardform usingonly integers. Sotution Sincewe know the slopeandy-rntercept, startwith slope-intercept form: 11 - V : -x I "2J Slope-interceptform - - xl 1 * v : -/a LJ - 6 /( - t 1* * y\) : -6 ' I : . ) L J Multiply by -6 to getintegers. J 3x- 6y: -2 Standardform with integers Any integral multiple of 3x - 6y : -2 would also be standard form, but we usually use the smallestpossiblepositive coefficient for x. T ParattetLines Two lines in a plane are said to be parallel if they have no points in common. Any two vertical lines are parallel, and slope can be used to determine whether nonvertical lines are parallel. For example, the lines y : 3x - 4 and y : 3x f I are parallel becausetheir slopesare equal and theiry interceptsare different. 44 Chapter1 www Inequalities, Equations, and Modeling Theorem:ParallelLines Two nonvertical lines in the coordinate plane are parallel if and only if their ; slopes are equal. A proof to this theorem is outlined in Exercises97 and98. e&rr".f/t of paratteLtines tr Writing equatiCIns Find the equationin slope-interceptform of the line through (1, -4) that is parallel toy:3xt2. Sohltion Since y : 3x * 2 has slope 3, any line parallel to it also has slope 3. Write the equationof the line through ( 1, -4) with slope 3 in point-slope form: l0 Y-(-4):3(xy+4:3x-3 -10 !:3x-7 llEl 1) Point-slopeform Slope-interceptform E'tl _10 Figure1.35 Theline y : 3x - 7 goesthrough(1, -4) andis parallelto y : 3x -f 2. ru The graphsof !1 : 3x 7 and y2: 3x + 2 in Fig. 1.35 supportthe I answer. PerpendicutarLines Two lines are perpendicular if they intersectat a right angle. Slope can be used to determine whether lines are perpendicular. For example, lines with slopes such as 213 and -312 are perpendicular.The slope -312 is the opposite of the reciprocal of 213. In the following theorem we use the equivalent condition that the product of the slopes of two perpendicular lines is -1, provided they both have slopes. Theorem: PerpendicularLines i l|*o mes wi-lhslon.*--: d mz oft ?atpa'ndi,cular,if and onV ,rt:,**,r: - l. Figure1.36 PROOF The phrase "if and only if" means that there are two statementsto prove. First we prove that if /1 with slope m1 and 12with slope tn2 zre perpendicular, then rltrltt2 - - l. Assume that m1 > 0. At the intersectionof the lines draw a right triangle using a rise of ml and a run of 1, as shown in Fig 1.36.Rotate /1 (along with the right triangle) 90 degrees so that it coincides with /2. Now use the triangle in its new position to determine thatm2 : + ot ffi!fl2 - - 1. that the The secondstatementto prove is that lflfli2 - - 1 or fti2 : +implies perpendicular. lines are Start with the two intersecting lines and the two congruent right triangles,as shown in Fig. 1.36.It takesa rotation of 90 degreesto get the vertical sidemarkedmlIo coincidewith the horizontal sidemarkedmr Sincethe triangles are congruent, rotating 90 degreesmakes the triangles coincide and the lines coincide. So the lines are perpendicular. 1.4 wst' e-*r""/. LinearEquations in TwoVariables writing equatitrn$ sf, petrpffindicerl"ar tinss Z Find the equation of the line perpendicularto the line 3x - 4y : g and containing the point (-2,I). Write the answerin slope-interceptform. $otrt:tiCIn Rewrite 3x - 4y : 8 in slope-interceptform; lY:-3x+8 ., "v4: - x J - 2 S l o p eo f t h i s l i n e i s 3 / 4 . Sincethe product of the slopesof perpendicularlines is - 1, the slope of the line that we seekis -413. Use the slope -413 and the point (-2,l) in the point-slopeform: 4 y-l:--(x-(-2)) 3 v - 10 , 4-9 - - x | - a JJ 45 v - --x - - J ' t a JJ -15.2 The last equation is the required equation in slope-interceptform. The graphs of thesetwo equationsshould look perpendicular. ru -10 Ifyougraph lr:1. - 2and!2: -+, - withagraphingcalculator,the f graphs will not appearperpendicularin the standardviewing window becauseeach axis has a different unit length. The graphs appearperpendicular in Fig. I .37 becausethe unit lengthswere made equal with the ZSquarefeatureof the TI-83. I Figure'1.37 Apptications In Section 1.3 the distanceformula was usedto establishsome facts about geometric figures in the coordinateplane. We can also use slope to prove that lines are parallel or perpendicularin geometric figures. A**r"'fl, E | 2 3 4 B(5,0)7 8 x The diagonatscf a rhombus are perpendicular Given a rhombus with vertices (0, 0), (5, 0), (3,4), and (8, 4), prove that the diagonals of this rhombus are perpendicular.(A rhombus is a four-sided figure in which the sideshave equal length.) $ottltisn Figure1.38 Plot the four points A, B, C, andD as shown in Fig. 1.38.You should show that each side of this figure has length 5, verifying that the figure is a rhombus. Find the slopes of the diagonalsand their product: |TIAC : 4-0 8-0 lTllC' 0-4 tTlBD : |TIBD : l. ;(-2) : -l 46 ChaPter 1 ffi,,ii E q u a t i o n sI,n e q u a l i t i e sa ,n d M o d e l i n g Since the product of the slopes of the diagonals is -1, the diagonals are perpenI dicular. If the value of one variable can be determined from the value of anothervariable, then we say that the first variable is a function of the secondvariable.Because the area of a circle can be determinedfrom the radius by the formula A : nr2, we say that A is a function of r. If y is determined from x by using the slope-intercept form of the equation of a line y : mx * b, then y is a linear function of x. The formula F : ?C + 32 expressesF as alinear function of C. We will discussfunctions in depth in Chapter 2. A"*r""fe tr A ffinearfnrnction From ABC Wirelessthe monthly cost for a cell phone with 100 minutes per month is $35, or 200 minutesper month for $45. SeeFig. 1.39.The cost in dollars is a linear function of the time in minutes. a. Find the formula for C. b. What is the cost for 400 minutes per month? 100 200 300 400 Time (in minutes) 500 F i g u r e1 . 3 9 Sclution a. Firstfind the slope: cz-cr m: t2 - ^ : 45-35 : l0 : o'lo 2oo- loo roo The slopeis $0.10per minute.Now find b by usingC:35,t: form C : mt I b: m : 0.10in theslope-intercept 35:0.10(100)+b 35:10+b 25:b So the formula is C 100, and 1.4 swri'ii LinearEquations in TwoVariables 47 b. Uset : 400inthe formulaC : 0.10t+ 25: C : 0 . 1 0 ( 4 0 0 )+ 2 5 : 6 5 r Thecostfor 400minutesper monthis $65. Note that in Example 9 we could have used the point-slope form C - C r : m(t - t) to gettheformul aC :.0.10r + 25. Tryi t. The situation in the next example leadsnaturally to an equation in standardform. A-*r",/t @ Interpreting stope A manager for a country market will spend a total of $80 on apples at $0.25 each and pearsat $0.50 each.Write the number of applesshe can buy as a linear function of the number of pears.Find the slope and interpret your answer. Solution Let a representthe number of applesand p representthe number of pears.Write an equation in standardform about the total amount spent: 0.25a+0.50p:80 r 025a :::;::: S.'vef.ra The equationa : 320 - 2p or a : -2p + 320 expressesthe number of applesas a function of the number of pears.Sincep is the first coordinateand a is the second the slope is -2 applesper pear. So if the number of apples is decreasedby 2, then the number of pearscan be increasedby I and the total is still $80. This makessense I becausethe pearscost twice as much as the apples. l I For Thought T r ue or F als e ?Ex p l a i n . 1. The slopeof the line through(2,2) and(3, 3) is 312.e 7. The slopeof the line y - 3 - 2x is 3. p 2. The slopeof the line through(-3, 1)and(-3,5)is 0. p 8. Every line in the coordinateplane has an equationin standardform. t 3. Any two distinct parallel lines have equal slopes.r 4. The graph of x : 3 in the coordinateplane is the single point (3, 0). r 5. Two lines with slopesm1andffi2&raperpendicularif tltt : -llm2.t 6. Every line in the coordinateplane has an equationin slope-interceptform. r' 9. The line y : 3x is parallel to the line y : -3x. F 10. The line x - 3y : 4 containsthe point ( l, - 1) and has slope t lZ. r 48 and Modeling lnequalities, Equations, Chapter1 *xm Find the slope of the line containing eachpair of points. (Example1) 23. Y : - 2 x + 4 4 v I ) t. (-2,3),(4,5)i ( - 1 , 2 ) ,( 3 ,6 ) I 3 . ( 1 , 3 ) ,( 3 , - 5 ) - 4 2 4. (2, -r), (5,-3) - ; 5 . (5 ,2 ) , ( - 3, 2) o 6. (o,o),(5,0)o lt r\ /t l\ t' (r' d'V't)' 8. J (r t\ - ^ r-1 | 2 / ' \ o' :/ 3 ' \ I J 10. ( - 7 , 2 ) ,( - 7 , - 6 ) No slope 9. (5, - l), (5, 3) No slope Find the equation of the line through the given pair of points. Solve it for y if possible. (Example2) 1 1 .( - 1 ,- l ) , ( 3 , 4y) : 1 . . : '7 13. (-2.6). (4. -l) 'Y- 6^' 1 5 . ( 3 , 5 ) ,( - 3 , 5 ) y : s 17. (4, -3), (4, 12) x : a ll 3 (3,5)y ::, . + 12.(-2,1), : -1" . 14.(-3,5),(2,I)y + 1 6 . ( - 6 , 4 ) , ( 2 , 4 )y : + 1 8 . ( - 5 , 6 ) , ( - 5 , 4 )x : - s Write an equation in slope-interceptformfor each of the lines 2 and 3) shown. (Examples 1 19.y:'ir-t 20.Y:-xt2 Write each equation in slope-interceptform and identifi the slope and y-intercept of the line. (Example3) 27,3x - 5Y: l0 , : 1 * - 2 , 1(,0-,z ) "5) 2e.y-3-2@-a) y : 2 x - 5 , 2 , ( 0 ,- 5 ) t. 3r.y*1:t(*-(-3)) I I I .r) v: t* - t,t' r\"'tl 3 3 .y - 4 : 0 y : 4 , 0 ,( 0 , 4 ) 35. y - 0.4: 2 1 .v : 53 _-r2n'z 2 2 .y : 4 4 -1x +- J 28. 2x - 2Y : I y: x ),t,,(0,-tl2) y + 5: -3(x - (-l)) -3, (0,-8) ! : -3x - 8, 3. y-2:-;(x+5) 3 ( -t l l \ lt - 3 y -;- - ;. -t'\.0' 34. - y - 1 5 : 0 y : 5 , 0 ,( 0 ,5 ) ) -2.6) 0 . 0 3 ( x - 1 0 0 )y : 0 . 0 3 x- 2 . 6 , 0 . 0 3( ,0 , -0.26) 36. y + 0.2 : 0.02(x - 3) y : 0.02x- 0.26,0.02'(0, Use the y-intercept and slope to sketch the graph of each equation. (Example4) T 37.y:I"-, 3 9 .Y : -3x * 1 4r.y: -1* - t ) 38. .v : : x * 1 a J 4 0 .v : - x * 3 a J 4 2 .y = - ; x z tDue to spaceconstrictions,answersto theseexercisesmay be found in the complete Answers beginning on page A-l in the back of the book' 1.4 mreg+; Exercises 43.x-!:3 4 4 .2 x - 3 y : 6 Find the value of a in each case. (Examplesl 7) 45.y-5:0 4 6 .6 - y : 0 75. The line through (-2,3) Find the equation of the line through the given pair of points in standardform using only integers. See the strategyfor finding the equation of a line on page 43. (Example5) 47. (3,0) and (0, -4) 4x-3Y:12 4 9 . ( 2 , 3 ) a n d( - 3 , - l ) 4x-5Y:-7 5 1 . ( - 4 , 2 ) a n d( - 4 , 5 ) x : - 4 s s(. 2 , ) , ( - ),-r) 48. (-2,0) and (0, 3) 3x-2Y:-6 50. (4, -l) and(-2, -6) 5 x- 6 Y : 2 6 5 2 . ( - 3 , 6 ) a n d( 9 , 6 ) y : 6 '0.G,-,),(-',) l6x - l5Y: )/ 100x+184y:-477 49 and (8, 5) is perpendicularto Y:Qx+2.-5 76. The line through (3, 4) and (7, a) has slope213. ZOIZ 77. The line through (-2, a) and (a, 3) has slope - | 12. S 78. The,line through (-1, a) and (3, -4) is parallel to y : ax. Either prove or disprove each statement. (Jsea graph only as a guide. Yourproof should rely on algebraic calculations. (Exampte8) 79. The points (-I,2), (2, -I), (3, 3), and (-2, -2) arethe vertices of a parallelogram.T 'u(-;,1),(;,-) ".(;,tJ,(-+,+) 80. The points (- 1, 1), (-2, -5), (2, -4), and (3, 2) arethe vertices of a parallelogram.T Find the slope of each line described. (Examples6 and 7) 81. The points (-5, - 1), (-3, -4), (3,0), and (1, 3) arethe vertices of a rectangle.T 20x+42Y:3 3x-50Y:-ll 57. Alineparallelto y : 0.5x - 9 0.s 82. The points (-5, - 1), (1, -4), (4,2), and (- 1, 5) arethe vertices ofa square.F 58. A line parallel to 3x - 9y : 4 t lz 59. Alineperpendicularto3y - 3x: 83. The points (-5, 1), (-2, -3), and(4,2) arethe verticesof a right triangle. r J -l 60. A line perpendicularto2x - 3y : 8 -zlz 84. The points (-4, -3), (1, - 2),(2,3),and (-3, 2) arethevertices of a rhombus. T 61. A line perpendicularto the line x : 4 0 62. Alineparallel toy : 5 6 ru (Jsea graphing calculator to solve eachproblem. Write an equation in standardform using only integersfor each of the lines described.In each case make a sketch. (Examples6 anclT) ( x - 5 ) l 3 a n d y 2 : x - 0 . 6 7 ( x+ 4 . 2 ) . D o t h e 8 5 . G r a p hl t : lines appearto be parallel?Are the lines parallel?yes, no 63. The line with slope 2, going through (1, -2) 2x - y : 4 86. Graph y1 : 99x andy2 : -x199. Do the lines appearto be perpendicular?Should they appearperpendicular?yes, yes 64. The line with slope -3, going through (-3,4) 3x + y : -5 65. The line through(1,4),parallel to y : -3x3x + y : 7 6 6 . T h e l i n e t h r o u g h ( - 2 , 3 ) p a r a l l e lt o y : ) x 67. The line parallel to 5x - 7y : 5x-7Y:)3 68. The line parallelto 4x * 9y : 4x+9Y:2 69. The line perpendicularto y : 3x+2Y:g 70. The line perpendicular to y : xt9Y:41 71. The line perpendicularto x 2x-tY:-5 72. The line perpendicular to 3x x-t3Y:g 73. The line perpendicularto.tr : + 6x - 2y: -g 35 and containing (6, 1) 5 and containing (-4,2) (2, -3) lx + S and containing 9x * 5 and containing (5, 4) 2y : 3 and containing (-3, 1) y : 9 and containing (0, 0) 4 and containing (2,5) y : 5 74. The line perpendicularto y : 9 and containing ( - 1, 3) x: -l (x3 - 8)l@2 -t 2x -r 4).UseTRACEtoexamine 87. Graphy: points on the graph. Write a linear function for the graph. Explain why this function is linear. y : x - 2,x3 - 8 - (x - 2)(x2+ 2x + 4) (x3 + 2x2 - 5x - 6)l@'t x - 6).Use 8 8 . G r a p hy : TRACE to examine points on the graph. Write a linear function for the graph. Factor x3 + 2x2 - 5x - 6 completely. y : x * l , ( x + l ) ( x + 3 ) ( x- 2 ) Solve each problem. (Examples9 and 10) 89. Celsius to Fahrenheit Formula Fahrenheit temperatureF is a linear function of Celsius temperature C. The ordered pair (0,32) is an orderedpair of this function because0"C is equivalent to 32F, the freezing point of water. The ordered pair (I00,2I2) is also an orderedpair of this function because 100'C is equivalentto 2l2oF, the boiling point of water. Use Chapter 1 +;; 50 E q u a t i o n Isn, e q u a l i t i easn,d M o d e l i n g the two given points and the point-slope formula to write F as a function of C. Find the Fahrenheittemperatureof an oven at '). r Js V n o\ .a f( _ - t l .- l 0 l " | 5 90. Cost o.fBtrsinessCards SpeedyPrinting charges$23 for 200 deluxebusinesscardsand $35 for 500 deluxebusinesscards. Given that the cost is a linear function of the number of cards printed find a formula for that function and find the cost of - 15.S-13 700 businesscards.C' ., 0.0-1lr 91. VolumeDiscount Mona Kalini givesa walking tour of Honolulu to one personfor $49.To increaseher business,she advertisedat the National Orthodontist Conventionthat she would lower the price by $ 1 per personfor each additional person.Write the costper personc as a function of the numberof people n on the tour. How much does she make for a tour with 40 people? t' - -50 rr.$4(X) 92. TicketPricing At $10 per ticket,Willie Williams and the Wranglerswill fill all 8000 seatsin the AssemblyCenter.The managerknows that for every $l increasein the price, 500 ticketswill go unsold.Write the numberof ticketssold n, as a function of the ticket price, p. How much money will be taken in if the ticketsare $20 each?r =. --500/;+ 13,000, 560.000 93. Lindberghs Air Speed CharlesLindbergh estimatedthat at the start of his historic flight the practical economicalair speedwas 95 mph and at 4000 statutemiles from the starting point it was 75 mph (www.charleslindbergh.com). Assume that the practicaleconomicalair speedS is a linear function of the distanceD from the startingpoint as shown in the accompanying figure. Find a formula for that function. .\0.0051J-9-5 New York 1 1 0 0m i Atlantic Ocean Figure for Exercise 94 95. Computersand Printers An office managerwill spenda total of $60,000 on computersat $2000 each and printers at $ I 500 each.Write the number of computerspurchasedas a function of the number of printers purchased.Find and interpret the slope. (. : _,lp + 30. incrcases by 4 thenr. ,1.I1.7., e6.Carpenters andaurynrJBecausel:1';'.n:f*Jn.o u "u.ry, contractorwill distribute a total of $2400 in bonusesto 9 carpentersand 3 helpers.The carpentersall get the sameamount and the helpersall get the sameamount. Write the amount of a helper'sbonus as a function of the amount of a carpenter's bonus.Find and interpretthe slope. It -. -3t + 800" 3. lf t'incrcitscs by lBl.then/r clecreases by 53. For Writing/Discussion 9 7 . Equal Slopes Show that if y : mx -t bl and y : mx * b2 are equationsof lines with equalslopes,but b1 * b2,then they have no point in common. 98. Unequal Slopes Show that the lines y : tntx * bl and ! : rftzx * b2 intersectat a point with x-coordinate (bz - b)l@r - mz.)providedm1 * m2.Explainhowthis exerciseand the previous exerciseprove the theoremthat two nonvertical lines are parallel if and only if they have equal slopes. E 99. SummingAngles Considerthe anglesshown in the accompanying figure. Show that the degreemeasureof angleI plus the degreemeasureof angle B is equal to the degreemeasure of angle C. 3so )' 12Jz 234 Distance(thousandsof miles) Figure for Exercise 93 2 I 94. SpeedOver Newfoundland In Lindbergh's flying log he recordedhis air speedoverNewfoundlandas 98 mph, I 100miles into his flight. According to the formula from Exercise93, what should have been his air speedover Newfoundland? 89.-5 mph -'7 {' .'i' ^l 0 Figure for Exercise99