f )

s)

I

I\*ir

I

1.4 ' t i l , , Linear Equations in Two Variables 39

Replace W and A with your actual weight and height. Now draw a graph show-ing how B depends on height for suitable values of H.

For a fixed weight and age, does the basic energy requirement increase or decreasefor taller persons? Increases

For the three graphs that you have drawn, explain how to determine whether thegraph is increasing or decreasing from the formula for the graph.

h)

1x2, ! t)

Figure 1.27

Defini t ion: Slope

In Section 1.3 we graphed lines, including horizontal and vertical lines. We learnedthat every line has an equation in standard form Ax * By : C. In this section wewill continue to study lines.

Slope of a lineA road that has a 5o/o grade rises 5 feet for every horizontal run of 100 feet. A roofthat has a 5-12 pitch rises 5 feet for every horizontal run of 12 feet. The grade of aroad and the pitch of a roof are measurements of steepness. The steepness or slope ofa line in the xy-coordinate system is the ratio of the rise (the change in y-coordinates)to the run (the change in x-coordinates) between two points on the line:

slooe _ change inl-coordinates _ rise' change in r-coordinates mn

If (x1, y1) and (*2, y) are the coordinates of the two points in Fig. l.27,then the risers lz

- y1 and the run is x2 - x1:

Theslopeofthe1inethrough(xr, ,1J'1"z,yz)withx1*x2is

l z - l ri

x z - x t

Note that if (x1 , y) and (*t, yr) are two points for which xl : x2 then the line throughthem is a vertical line. Since this case is not included in the definition of slope, a verti-cal line does not have a slope. We also say that the slope of a vertical line is undefined.If we choose two points on a horizontal line then y1 : 12dnd lz - lt : 0. For anyhorizontal line the rise between two points is 0 and the slope is 0.

40 Chapter 1 ruffiffi Equations, Inequali t ies, and Modeling

Figure 1.28

Theorem:Point-Slope Form

e**rr./t E Finding the stopeIn each case find the slope of the line that contains the two given points.

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 the formula:

. r l - V r - 2 - 4 - 6s l o p e : ' r ; :

_ l _ ( _ 3 ) :

2 - - 3

The slope of the line is -3.

b. Use (*v y) : ( -3, 7) and (rr, yr) : (5, 7) in the formula:

. v t - l r 7 - 7 0s lope : - ; = :5_ (_ " r : 8 :o

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

The slope of a line is the same number regardless of which two points on theline are used in the calculation of the slope. To understand why, consider the two tri-angles shown in Fig. 1.28. These triangles have the same shape and are called simi-lar triangles. Because the ratios of corresponding sides of similar triangles are equal,the ratio of rise to run is the same for either triangle.

Point-Stope FormSuppose that a line through (xr, y1) has slope m. Every other point (r, y) on the linemust satisff the equation

= : *

because any two points can be used to find the slope. Multiply both sides by x - x1to get ! - lr : m(x - xr), which is the point-slope form of the equation of a line.

In Section 1.3 we started with the equation of a line and graphed the line. Usingthe point-slope form, we can start with a graph of a line or a description of the lineand write the equation for the line.

lxq, y q)

(xs, y)

( x v y )

1.4 w m tur Linear Equations in Two Variables 41

e-*r"^fb A The equation of a tine given two pointsIn each case graph the line through the given pair of points. Then find the equationof the line and solve itfor y if possible.

^ . ( - 1 , 4 ) , ( 2 , 3 ) b . ( 2 , 5 ) , ( - 6 , 5 ) c . ( 3 , - 1 ) , ( 3 , 9 )

$olutisn

a. Find the slope of the line shown in Fig. t.29 as follows:

! z - ! t 3 - 4 - l 1" ' - * r - x 1 2 - ( - l ) 3 3

Now use a point, say (2,3), and m - -{ in ttt. point-slope form:

! - l r : m ( x - x r )L

y - 3 - -; (x - 2) The equation in point-slope form

; 2y - 3 : -r, + iI l l

i* *

T The equation solved fory

b. The slope of the line through (2,5) and (-6, 5)equation of this horizontal line is y : 5.

c. The line through (3, - 1) and (3, 9) shown in Fig.have slope. Its equation isx : 3.

shown in Fig. 1.30 is 0. The

1.31 is vertical and it does notI

Stope-Intercept FormTheline y : mx * bgoesthrough (0,b) and (7,m t b). Betweenthesetwopointsthe rise is m and the run is 1. So the slope is ru. Since (0, b) is the y-intercept and mis the slope, y : mx * b is called slope-intercept form. Any equation in standardform Ax * By : C canbe rewritten in slope-intercept form by solving the equationfory provided B + 0.lf B : 0, then the line is vertical and has no slope.

IIi

Figure 1 .31

Theorem:Slope-lntercept Form i*m;;ry.*ffffi;;ry'i

i : '!lr1..",,, -

Every nonvertical line has an equation in slope-intercept form.

If you know the slope and y-intercept for a line, then you can use slope-interceptform to write its equation. For example, the equation of the line through (0, 9) with

2 4 6 8 1 0

Figure 1.29

-:7 4 -5 -4 -3 -2

Chapter 1 f f iwi i# Equations, Inequali t ies, and Modeling

Figure 1.32

Figure 1.33

slope 4 is y - 4x * 9. In the next example we use the slope-intercept form to deter-mine the slope andy-intercept for a line.

Ar*r""N, E Find the stope and y-interceptIdenti$r the slope andy-intercept for the line 2x - 3y : 6.

$otution

First solve the equation fory to get it in slope-intercept form:

2 x - 3 y : 6-31,y : -2x -t 6

2! : 1x -2

So the slope is ] and they-intercept is (0, -2).

Using Stope to Graph a lineSlope is the ratio ffi that results from moving from one point to another on a line. Apositive rise indicates a motion upward and a negative rise indicates a motion down-ward. A positive run indicates a motion to the right and a negative run indicatesa motion to the left. If you start at any point on a line with slope ), thenmoving up1 unit and2 units to the right will bring you back to the line. On a line with slope -3

ot t',moving down 3 units and 1 unit to the right will bring you back to the line.

I

Graphs of y=vv1;s

, Figure 1.34

e**r"./t A Graphing a tine using its stopeand y-intercept

21 x + 4

Graph each line.

a . y : 3 x - I b .

$olution

a. The liney:3x - t hasy-intercept (0, -1)andslope: or J. Startingat (0, -1)

we obtain a second point on the line by moving up 3 units and I unit to the right.So the line goes through (0, - 1) and (1,2) as shown inFig. 1.32.

.b. The \ ine y: - ! , * 4 hasy- intercept (0,4) and slope -?otf . Start ingat

(0, 4) we obtain a second point on the line by moving down 2 units and then 3 unitsto the right. So the line goes through (0, 4) and (3, 2) as shown in Fig. 1.33. I

As the .r-coordinate increases on a line with positive slope, the y-coordinateincreases also. As the x-coordinate increases on a line with negative slope, they-coordinate decreases. Figure 1.34 shows some lines of the form y : mx with posi-tive slopes and negative slopes. Observe the effect that the slope has on the positionof the line.

t;L-l,

1.4 ff iff i i4rni Linear Equations in Two Variables

The Three Forms for the Equation of a LineThere are three forms for the equation of a line. The following strategy will help youdecide when and how to use these forms.

STRATEGY Finding the Equation of a Line

1 .

2.

3.

4.

5.

A-*rr"/t tr $tandard form using integersFind the equation of the line throurn (O,J) *i.n slope ]. Wrlt. the equation instandard form using only integers.

Sotution

Since we know the slope andy-rntercept, start with slope-intercept form:

1 1V : - x I - Slope-interceptform" 2 J

l 1- -x * v : -- / aL J

I-6 ' : Mul t ip ly by -6 to get integers.. ) L J

J

-2 Standard form with integers

Any integral multiple of 3x - 6y : -2 would also be standardally use the smallest possible positive coefficient for x.

Parattet LinesTwo lines in a plane are said to be parallel if they have no points in common. Anytwo vertical lines are parallel, and slope can be used to determine whether nonverti-cal lines are parallel. For example, the lines y : 3x - 4 and y : 3x f I are paral-lel because their slopes are equal and theiry intercepts are different.

/ 1 \-6 ( - t * * y ) :

3 x - 6 y :

form, but we usu-T

44 Chapter 1 www Equations, Inequali t ies, and Modeling

Theorem: Parallel Lines

l 0

llEl E ' t l

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 Exercises 97 and98.

e&rr".f/t tr Writing equatiCIns of paratteL tinesFind the equation in slope-intercept form of the line through (1, -4) that is parallelt o y : 3 x t 2 .

Sohltion

Since y : 3x * 2 has slope 3, any line parallel to it also has slope 3. Write theequation of the line through ( 1, -4) with slope 3 in point-slope form:

- 1 0

Y - ( - 4 ) : 3 ( x -

y + 4 : 3 x - 3

! : 3 x - 7

1) Point-slope form

Slope-intercept form

_ 1 0

Figure 1.35

Theorem:Perpendicular Lines

The line y : 3x - 7 goes through (1, -4) and is parallel to y : 3x -f 2.ru The graphs of !1 : 3x - 7 and y2: 3x + 2 in Fig. 1.35 supportanswer.

Perpendicutar LinesTwo lines are perpendicular if they intersect at a right angle. Slope can be usedto determine whether lines are perpendicular. For example, lines with slopessuch as 213 and -312 are perpendicular. The slope -312 is the opposite of thereciprocal of 213. In the following theorem we use the equivalent condition thatthe product of the slopes of two perpendicular lines is -1, provided they bothhave slopes.

i l|*o mes wi-lh slon.*--: d mz oft ?atpa'ndi,cular,if and onV ,rt:,**,r:

PROOF The phrase "if and only if" means that there are two statements to prove.First we prove that if /1 with slope m1 and 12 with slope tn2 zre perpendicular, thenrltrltt2 - - l. Assume that m1 > 0. At the intersection of the lines draw a right trian-gle using a rise of ml and a run of 1, as shown in Fig 1.36. Rotate /1 (along with theright triangle) 90 degrees so that it coincides with /2. Now use the triangle in its

new position to determine that m2 : + ot ffi!fl2 - - 1.

The second statement to prove is that lflfli2 - - 1 or fti2 : +implies that thelines are perpendicular. Start with the two intersecting lines and the two congruentright triangles, as shown in Fig. 1.36. It takes a rotation of 90 degrees to get the verti-cal side marked mlIo coincide with the horizontal side markedmr Since the trianglesare congruent, rotating 90 degrees makes the triangles coincide and the lines coin-

theI

-

l.

Figure 1.36 cide. So the lines are perpendicular.

1.4 wst ' Linear Equations in Two Variables

e-*r""/. Z writing equatitrn$ sf, petrpffindicerl"ar tinssFind the equation of the line perpendicular to the line 3x - 4y : g and containingthe point (-2,I). Write the answer in slope-intercept form.

$otrt:t iCIn

Rewrite 3x - 4y : 8 in slope-intercept form;

l Y : - 3 x + 8

- 2 Slope of th is l ine is 3/4.

Since the product of the slopes of perpendicular lines is - 1, the slope of the line thatwe seek is -413. Use the slope -413 and the point (-2,l) in the point-slope form:

4y - l : - - ( x - ( - 2 ) )

3

, 4 9v - | - - - x - -a

J J

.,J

v : - x" 4

4 5v - - - x - -

J ' t a

J J

-15.2

- 1 0

Figure '1.37

The last equation is the required equation in slope-intercept form. The graphs ofthese two equations should look perpendicular.

ru Ifyougraph lr:1. - 2and!2: -+, - f withagraphingcalculator,the

graphs will not appear perpendicular in the standard viewing window because eachaxis has a different unit length. The graphs appear perpendicular in Fig. I .37 be-cause the unit lengths were made equal with the ZSquare feature of the TI-83. I

AppticationsIn Section 1.3 the distance formula was used to establish some facts about geometricfigures in the coordinate plane. We can also use slope to prove that lines are parallelor perpendicular in geometric figures.

A**r"'fl, E The diagonats cf a rhombusare perpendicular

Given a rhombus with vertices (0, 0), (5, 0), (3,4), and (8, 4), prove that the diago-nals of this rhombus are perpendicular. (A rhombus is a four-sided figure in whichthe sides have equal length.)

$ottltisn

Plot the four points A, B, C, andD as shown in Fig. 1.38. You should show that eachside of this figure has length 5, verifying that the figure is a rhombus. Find the slopesof the diagonals and their product:

10

| 2 3 4 B ( 5 , 0 ) 7 8 x

Figure 1.38

4 - 08 - 0

tTlBD :

l .

; ( - 2 ) :

0 - 4|TIAC :

lTl lC' |TIBD : - l

46 ChaPter 1 ffi,,ii Equat ions , Inequa l i t ies , and Mode l ing

Since the product of the slopes of the diagonals is -1, the diagonals are perpen-dicular. I

If the value of one variable can be determined from the value of another vari-able, then we say that the first variable is a function of the second variable. Becausethe area of a circle can be determined from the radius by the formula A : nr2, wesay that A is a function of r. If y is determined from x by using the slope-interceptform of the equation of a line y : mx * b, then y is a linear function of x. The

formula F : ?C + 32 expresses F as alinear function of C. We will discuss func-

tions in depth in Chapter 2.

A"*r""fe tr A ffinear fnrnctionFrom ABC Wireless the monthly cost for a cell phone with 100 minutes per month is$35, or 200 minutes per month for $45. See Fig. 1.39. The cost in dollars is a linearfunction 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 500Time (in minutes)

Figure 1.39

Sclution

a. First find the slope:

c z - c r 4 5 - 3 5 l 0m :

t2 - ^ :

2oo- loo :

roo : o' lo

The slope is $0.10 per minute. Now f ind b by using C:35,t:m : 0.10 in the slope-intercept form C : mt I b:

100, and

3 5 : 0 . 1 0 ( 1 0 0 ) + b

3 5 : 1 0 + b

2 5 : b

So the formula is C

1.4 swri ' i i Linear Equations in Two Variables

b. Use t : 400 inthe formula C : 0.10t + 25:

C : 0 . 1 0 ( 4 0 0 ) + 2 5 : 6 5

The cost for 400 minutes per month is $65.

47

r

r

Note that in Example 9 we could have used the point-slope formC - Cr : m(t - t ) to getthe formulaC : .0.10r + 25. Tryi t .

The situation in the next example leads naturally to an equation in standard form.

A-*r",/t @ Interpreting stopeA manager for a country market will spend a total of $80 on apples at $0.25 eachand pears at $0.50 each. Write the number of apples she can buy as a linear functionof the number of pears. Find the slope and interpret your answer.

Solution

Let a represent the number of apples and p represent the number of pears. Write anequation in standard form about the total amount spent:

0 . 2 5 a + 0 . 5 0 p : 8 0

025a

: : : ; : : : S . ' ve f . ra

The equation a : 320 - 2p or a : -2p + 320 expresses the number of apples asa function of the number of pears. Sincep is the first coordinate and a is the secondthe slope is -2 apples per pear. So if the number of apples is decreased by 2, thenthe number of pears can be increased by I and the total is still $80. This makes sensebecause the pears cost twice as much as the apples. I

lI

For ThoughtTrue or False? Explain.

1. The slope of the l ine through(2,2) and (3, 3) is 312. e

2. The slope of the l ine through (-3, 1)and (-3,5)is 0. p

3. Any two distinct parallel lines have equal slopes. r

4. The graph of x : 3 in the coordinate plane is the singlepoint (3, 0). r

5. Two lines with slopes m1andffi2&ra perpendicular if

t l t t : - l lm2. t

6. Every line in the coordinate plane has an equation inslope-intercept form. r'

7. The slope of the line y - 3 - 2x is 3. p

8. Every line in the coordinate plane has an equation instandard form. t

9. The line y : 3x is parallel to the line y : -3x. F

10. The line x - 3y : 4 contains the point ( l, - 1) and has

slope t lZ. r

48 Chapter 1 *xm Equations, lnequali t ies, and Modeling

Find the slope of the line containing each pair of points. (Example 1) Y : - 2 x + 423. 4v

I

t . ( -2,3), (4, 5) i3. (1 ,3 ) , (3 , -5 ) -4

5. (5, 2) , ( -3,2) o

l t r \ / t l \t ' ( r ' d 'V' t) '9. (5, - l ) , (5, 3) No slope

)

4.

6.

8.

10.

I- ^J

Find the equation of the line through the given pair of points. Solve

it for y if possible. (Example 2)

No slope

( -1,2) , (3 , 6) I

(2, -r), (5, -3) -

(o, o), (5,0) o

r-1 | (r t\\ 3 ' 2 / ' \ o ' : /

( -7 ,2) , ( -7 , -6)

2;J

11 . ( -1 , - l ) , ( 3 ,4 ) y : 1 . . :13 . ( - 2 .6 ) . ( 4 . - l ) ' 7 l l

' Y - 6 ^ ' 315 . ( 3 ,5 ) , ( - 3 , 5 ) y : s

17. (4, -3), (4, 12) x : a

12. (-2,1), (3, 5) y ::, . +14. (-3,5),(2, I)y : -1" . +16. ( -6 ,4 ) , (2 ,4 ) y : +

18. ( -5 ,6 ) , ( -5 , 4 ) x : -s

42 2 . y : - 1 x

Write an equation in slope-interceptformfor each of the lines

shown. (Examples 2 and 3)

1

1 9 . y : ' i r - t 2 0 . Y : - x t 2

Write each equation in slope-interceptform and identifi the slope

and y-intercept of the line. (Example 3)

27, 3x - 5Y: l0

, :1* - 2,1, (0, -z)" 5 )

2 e . y - 3 - 2 @ - a )y : 2x - 5 ,2 , (0 , -5)

t .3 r . y *1 : t ( * - ( - 3 ) )

I I I r . r )v: t* - t,t ' \" ' t l3 3 . y - 4 : 0

y : 4 ,0 , (0 ,4 )35. y - 0 .4 : 0 .03(x - 100) y : 0 .03x - 2 .6 ,0 .03, (0 , -2 .6)

36. y + 0.2 : 0.02(x - 3) y : 0.02x - 0.26,0.02' (0, -0.26)

Use the y-intercept and slope to sketch the graph of each equation.

(Example 4) T

28. 2x - 2Y: I

y : x - ) , t , , ( 0 , - t l 2 )

34.

y + 5 : - 3 ( x - ( - l ) )! : -3x - 8, -3, (0, -8)

3 .y - 2 : - ; ( x + 5 )

3 l t 3 ( l l \y - - ; - -

; . - t ' \ .0 ' - t

)- y - 1 5 : 0y : 5 ,0 , (0 , 5 )

2 1 . v :5 3_ - r -2 n ' z

4+ -

J

37.y: I " - ,3 9 . Y : - 3 x * 1

4r. y: -1* - t

38. .v :

4 0 . v :

4 2 . y =

): x * 1aJ

- x * 3

aJ- ; xz

tDue to space constrictions, answers to these exercises may be found in the complete Answers beginning on page A-l in the back of the book'

4 3 . x - ! : 3

4 5 . y - 5 : 0

47. (3,0) and (0, -4)4 x - 3 Y : 1 2

49 . ( 2 ,3 ) and ( -3 , - l )4 x - 5 Y : - 7

4 4 . 2 x - 3 y : 6

4 6 . 6 - y : 0

Find the equation of the line through the given pair of points instandard form using only integers. See the strategy for finding theequation of a line on page 43. (Example 5)

1.4 m re g+; Exercises 49

Find the value of a in each case. (Examples l 7)

75. The line through (-2,3) and (8, 5) is perpendicular to

Y : Q x + 2 . - 5

76. The line through (3, 4) and (7, a) has slope 213. 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. (Jse a graph only as aguide. Your proof should rely on algebraic calculations. (Exampte 8)

79. The points (-I,2), (2, -I), (3, 3), and (-2, -2) are the ver-tices of a parallelogram. T

80. The points (- 1, 1), (-2, -5), (2, -4), and (3, 2) are the ver-tices of a parallelogram. T

81. The points (-5, - 1), (-3, -4), (3,0), and (1, 3) are the ver-tices of a rectangle. T

82. The points (-5, - 1), (1, -4), (4,2), and (- 1, 5) are the ver-tices ofa square. F

83. The points (-5, 1), (-2, -3), and(4,2) are the vertices of aright triangle. r

84. The points (-4, -3), (1, - 2), (2,3),and (-3, 2) arethe ver-tices of a rhombus. T

ru (Jse a graphing calculator to solve each problem.

85 . G raph l t : ( x - 5 ) l 3andy2 : x - 0 .67 (x + 4 .2 ) .Do thelines appear to be parallel? Are the lines parallel? yes, no

86. Graph y1 : 99x and y2 : -x199. Do the lines appear to beperpendicular? Should they appear perpendicular? yes, yes

87. Graph y: (x3 - 8)l@2 -t 2x -r 4).UseTRACEtoexaminepoints on the graph. Write a linear function for the graph. Ex-plain why this function is linear.y : x - 2,x3 - 8 - (x - 2)(x2 + 2x + 4)

88. Graph y : (x3 + 2x2 - 5x - 6) l@' t x - 6) .UseTRACE to examine points on the graph. Write a linear func-tion for the graph. Factor x3 + 2x2 - 5x - 6 completely.y : x * l , (x + l ) (x + 3) (x - 2)

Solve each problem. (Examples 9 and 10)

89. Celsius to Fahrenheit Formula Fahrenheit temperature F is alinear function of Celsius temperature C. The ordered pair(0,32) is an ordered pair of this function because 0"C is equiv-alent to 32F, the freezing point of water. The ordered pair(I00,2I2) is also an ordered pair of this function because100'C is equivalentto 2l2oF, the boiling point of water. Use

48. (-2,0) and (0, 3)3 x - 2 Y : - 6

50. (4 , - l ) and ( -2 , -6)5 x - 6 Y : 2 6

51. ( -4 , 2) and ( -4 ,5) x : -4 52. ( -3 ,6) and (9 ,6) y : 6

ss . (2 , ) , ( - ) , - r )l 6 x - l 5Y : ) /

".(;,tJ,(-+,+)3 x - 5 0 Y : - l l

'0.G,-,),(-',)1 0 0 x + 1 8 4 y : - 4 7 7

'u (-;,1),(;,-)2 0 x + 4 2 Y : 3

Find the slope of each line described. (Examples 6 and 7)

57. Alineparallelto y : 0.5x - 9 0.s

58. A line parallel to 3x - 9y : 4 t lz

59. Al ineperpendicularto3y - 3x: J - l

60. A line perpendicular to2x - 3y : 8 -zlz

61. A line perpendicular to the line x : 4 0

62. Alineparallel toy : 5 6

Write an equation in standard form using only integers for each ofthe lines described. In each case make a sketch. (Examples 6 ancl T)

63. The line with slope 2, going through (1, -2) 2x - y : 4

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

66. Thel inethrough(-2 ,3)para l le l toy : )x + 6x - 2y : -g

67. The line parallel to 5x - 7y : 35 and containing (6, 1)5 x - 7 Y : ) 3

68. The line parallelto 4x * 9y : 5 and containing (-4,2)4 x + 9 Y : 2

69. The line perpendicular to y : lx + S and contain ing (2, -3)

3 x + 2 Y : g

70. The line perpendicular to y : 9x * 5 and containing (5, 4)x t 9 Y : 4 1

71. The line perpendicular to x - 2y : 3 and containing (-3, 1)2 x - t Y : - 5

72. The line perpendicular to 3x - y : 9 and containing (0, 0)x - t 3 Y : g

73. The line perpendicular to.tr : 4 and containing (2,5) y : 5

74. The line perpendicular to y : 9 and containing ( - 1, 3)x : - l

50 Chapter 1 +;; Equat ions, Inequal i t ies, and Model ing

the two given points and the point-slope formula to write F asa function of C. Find the Fahrenheit temperature of an oven atr s n o a - ' ) .r JV \ . f ( _ - t l . - l 0 l " |

5

90. Cost o.f Btrsiness Cards Speedy Printing charges $23 for 200deluxe business cards and $35 for 500 deluxe business cards.Given that the cost is a linear function of the number of cardsprinted find a formula for that function and find the cost of700 business cards. C' . , 0.0-1lr - 15. S-13

91. Volume Discount Mona Kalini gives a walking tour ofHonolulu to one person for $49. To increase her business, sheadvertised at the National Orthodontist Convention that shewould lower the price by $ 1 per person for each additional per-son. Write the cost per person c as a function of the number ofpeople n on the tour. How much does she make for a tour with40 people? t' - -50 rr. $4(X)

92. Ticket Pricing At $10 per ticket, Willie Williams and theWranglers will fill all 8000 seats in the Assembly Center. Themanager knows that for every $l increase in the price, 500tickets will go unsold. Write the number of tickets sold n, as afunction of the ticket price, p. How much money will be takenin if the tickets are $20 each? r =. --500/; + 13,000, 560.000

93. Lindbergh s Air Speed Charles Lindbergh estimated that atthe start of his historic flight the practical economical airspeed was 95 mph and at 4000 statute miles from the startingpoint it was 75 mph (www.charleslindbergh.com). Assumethat the practical economical air speed S is a linear functionof the distance D from the starting point as shown in the ac-companying figure. Find a formula for that function.. \ - 0 . 0 0 5 1 J - 9 - 5

99.

1 2 J zDistance (thousands of miles)

Figure for Exercise 93

Speed Over Newfoundland In Lindbergh's flying log herecorded his air speed over Newfoundland as 98 mph, I 100 milesinto his flight. According to the formula from Exercise 93,what should have been his air speed over Newfoundland?89.-5 mph

1100 miNew York

Atlantic Ocean

Figure for Exercise 94

95. Computers and Printers An office manager will spend atotal of $60,000 on computers at $2000 each and printers at$ I 500 each. Write the number of computers purchased as afunction of the number of printers purchased. Find and inter-pret the slope. (. : _, lp + 30. ,1. I1.7., incrcases by 4 then r.

e6. Carpenters and aurynrJ Becausel:1';'.n:f*Jn.o "u.ry,

ucontractor will distribute a total of $2400 in bonuses to 9 car-penters and 3 helpers. The carpenters all get the same amountand the helpers all get the same amount. Write the amount ofa helper's bonus as a function of the amount of a carpenter'sbonus. Find and interpret the slope.It - . -3t + 800" 3. l f t ' incrcitscs by lBl. then /r clecreases by 53.

For Writ ing/Discussion

97.

98.

Equal Slopes Show that if y : mx -t bl and y : mx * b2are equations of lines with equal slopes, but b1 * b2,thenthey have no point in common.

Unequal Slopes Show that the lines y : tntx * bl and

! : rftzx * b2 intersect at a point with x-coordinate(bz - b)l@r - mz.) provided m1 * m2. Explainhowthisexercise and the previous exercise prove the theorem that twononvertical lines are parallel if and only if they have equalslopes.

Summing Angles Consider the angles shown in the accom-panying figure. Show that the degree measure of angle I plusthe degree measure of angle B is equal to the degree measureof angle C.

)'2

I

{' . ' i '-'7^l

0

Figure for Exercise 99

E

3 s o

94.

2 3 4