Final Countdown - Manchester High School

______________________ Name

MATH 2020

Final Countdown

Unit 4

Pages Due

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Lesson 1

Lesson 2

Lesson 3

Quiz 1-3

Lesson 4

Lesson 5

Lesson 6

Lesson 7

Quiz 4-7 ___________________________

Key Vocabulary .. linear equation,

I p. 1

4

4 solution of a linear

equation, p. 1

44

An ordered pair (x, y) is used to locate a point in a coordinate plane.

Check

-3

-3

� J<ey ldea Linear Equations

A linear equation is an equation whose graph is a line. The points on

the line are solutions of the equation.

You can use a graph to show the solutions of a linear equation. The

graph below represents the equation y = x + 1.

X y (x, y)

l f, (-1, u)

0 I (0, 1) y-

2 3 (2, )

Graphy = -2x + I.

Step 1: Make a table of values.

X y = -2x + 1 y (x, y)

1 y= -2( 1) + 1 ' (-1, 3)

0 y = -2(0) + 1 1 (0, 1)

- y= -2(>) + l (2, -3)

y-

Step 2: Plot the ordered pairs.

Step 3: Draw a line through the points.

ldea Graphing Horizontal and Vertical Lines

The graph of y = bis a horizontal

line passing through (0, b).

The graph of x = a is a vertical

line passing through (a, 0).

)' )' x=a

y- b

(O, b) 0

(a, O)

X

0 X

144 Chapter 4 Graphing and Writing Linear Equations .. Multi-language Glossary cit BigidecisMcith�om

EXAMPLE

a. Graph y = -3.

The graph of y = -3 is ahorizontal line passing through(0, -3). Draw a horizontal line

through this point.

--,- rl I 1-3-2

b. Graph x = 2.

The graph of x = 2 is avertical line passing through(2, 0). Draw a vertical line

through this point.

y 2-----.....

1 (2, 0)

-2

-, On Your Own

EXAMPLE ID

A tropical storm becomes a hurricane when wind speeds are at least 74 miles per hour.

Graph the linear equation. Use a graphing calculator to check your

graph, if possible.

1. y = 3x 2. y = -2x + 2 3. x = -4 4. y = -1.5

The wind speedy (in miles per hour) of a tropical storm is

y = 2x + 66, where xis the number of hours after the storm

enters the Gulf of Mexico.

a. Graph the equation.

b. When does the storm become a hurricane?

a. Make a table of values.

X y = 2x + 66

0 y = 2(0) ,- 66

1 y = 2(1) + 66

2 y=2(2)+66

3 y = 2(3) + 66

y (x, y)

G6 (0, 66)

68 (1, 68)

70 (2, 70)

72 (3, 72)

Plot the ordered pairs and draw a line through the points.

b. From the graph, you can see thaty = 74 when x = 4. So, the stormbecomes a hurricane 4 hours after it enters the Gulf of Mexico.

• On Your Own5. WHAT IF? The wind speed of the storm is y = l.5x + 62.

When does the storm become a hurricane?

Section 4.1 Graphing Linear Equations 145

Key Vocabulary ..

slope, p. 150

rise, p. 150

run, p. 150

In the slope formula, x1 is read as "x sub one," and y

2 is read as "y sub

two." The numbers 1 and 2 in x1 and y

2 are

called subscripts.

Slope

The slope m of a line is a ratio of the change in y (the rise) to the change in x (the run) ben,veen any two points, (x

1, y

1) and (x

2, y

2), on the line.

m=

rise = changt'.! in y =

)'2 - Y1

run change in x x, x1

Positive Slope

The line rises from left to right.

X

Negative Slope

)'

X

The line falls from left to right.

�-:=r;-:::m:::::-:�-::::;;:=-=-��;=-::fi�;:-::.�.-;-:-; .. �;-:----:--��;,;.-;,� EXAMPLE 1l.-.;:,:==:.-..:=.:=.r=.:::::.:=-:=:::::.._ ......... ....-_........_ __ ".,_.,- ;.,;...�

When finding slope, you can label either point as (x1, y1) and the other point as (x2, Y/

Describe the slope of the line. Then find the slope.

a. r---r----, b. .v

The line rises from left to right. So, the slope is positive. Let (x

i' y

1) = (-3, -1) and

Cx2, Y2) = (3, 4).

4 - (-1) =---

3 - (-3)

5

6

3

-4 -3 -2 - I 2 3 4 X

S (1,-2)

=:� -4

I The line falls from left to right. So, the slope is negative. Let (x

i' y

1) = (-1, 1) and

(X2 , Y2) = (1, -2). V V

m = .:..1.__:_!_ X,, -x

1

2 l =---

1 ( l) 3 3

=- or--2 ' 2

150 Chapter 4 Graphing and Writing Linear Equations .. Multi-lan9uage Glossary at BigideasMa+h�om

ii On Your Own

�+9u',;e �ady. Exercises 7-9

EXAMPLE

EXAMPLE

The slope of every horizontal line is 0. The slope of every vertical line is undefined.

�-Xo.u 're &a.ct }' Exercises 13-15

Find the slope of the line.

1. 2. 1--t--+-+ 5 +------+--1------4...---<

(3, 2)

-3 -2 -I I 2 3 X

7


-4

(-4, -1)_2t---1--+--,---3·>--+--+---<

3.

5-56 (-1)

2 +-->---,--+---+--+--+---<

J +-1------l--+---i--+---+

= Q orO7' -2-i 1 2 3 4 5 6 X

::• The slope is 0.


=6-2

4-4

5

4

3

2

l·

-]

(4, 2)

l 2 3 5 X

::• Because division by zero is undefined, the slope of the line is undefined.

-, On Your Own

Find the slope of the line through the given points.

4. (1,-2),(7,-2) 5. (-2,4),(3,4)

6. (-3, -3), (-3, -5) 7. (0, 8), (0, 0)

y

8. How do you know that the slope of every horizontal line is O? Howdo you know that the slope of every vertical line is undefined?

I x

Section 4.2 Slope of a Line 151

152

EXAMPLE

� 'touJ;�ady Exercises 21-24

The points in the table lie on a line. How can you find the slope of the

line from the table1What is the slope?

I: I 1 4 7 10

8 6 4 2

Choose any two points from the table and use the slope formula.

Use the points (x 1 , y1 ) = (1, 8) and (x2

, y2

) = (4, 6).

6 8

-2 2= - or--] ' 3

::• The slope is -�.3

� On Your Own

Check

6

5

4 1-----+---------+-

3 --------

2 t---J�-+--+--i----t---:----:c+-:"�

�It--r--t--+-+-t---��r--1

1 2 3 4 5 6 7 8 9 lOx

The points in the table lie on a line. Find the slope of the line.

•. I : I � I : I : I 1

7

1 I 10

I : I -63

I �2

I �

I

I : I

') Summary Slope

Positive Slope Negative Slope SlopeofO

.v

Undefined Slope

.v

The line rises from left to right.

The line falls from left to right.

0

The line is horizontal.

X 0

The line is vertical.

X

Chapter 4 Graphing and Writing Linear Equations

equation y = mx, m

represents the constant of proportionality, the slope, and the unit rate.

-"'

0 �

EXAMPLE

Internet Plan

8 10

OQ 1 2 3 4 5 6 X

Data used (gigabytes)

EXAMPLE �

In Example 2, the slope indicates that the weight of an object on Titan is one-seventh its weight on Earth.

™100• Lesson Tutorials

8i9IdeasMatn

') l<ey Idea Direct Variation

Words When two quantities x andy are proportional, the relationship can be represented by the direct variation equation y = mx, where m is the constant of proportionality.

Graph The graph of y = mx is a line with a slope of m that passes through the origin.

y

The costy (in dollars) for x gigabytes of data on an Internet plan is

represented byy = lOx. Graph the equation and interpret the slope.

The equation shows that the slope mis 10. So, the graph passes through (0, 0) and (1, 10).

Plot the points and draw a line through the points. Because negative values of x do not make sense in this context, graph in the first quadrant only.

::• The slope indicates that the unit cost is $10 per gigabyte.

The weighty of an object on Titan, one of Saturn's moons, is proportional to the weight x of the object on Earth. An object that

weighs 105 pounds on Earth would weigh 15 pounds on Titan.

a. Write an equation that represents the situation.

Use the point (105, 15) to find the slope of the line.

y=mx

15 = m(105)

1 -=m

Direct variation equation

Substitute 15 for y and 105 for x.

Simplify.

•• So, an equation that represents the situation is y = .!.x.:•· 7

b. How much would a chunk of ice that weighs 3.5 pounds on Titan

weigh on Earth?

3.5 = .!. x Substitute 3.5 for y. 7

24.5 = X Multiply each side by 7.

::• So, the chunk of ice vvould weigh 24.5 pounds on Earth.

160 Chapter 4 Graphing and Writing Linear Equations

X

i, On Your Own

-- y� 12 * 10§ 8� 6

� 4

0 2

�ow Y.ou're Re d a. :yExercises 7-8

EXAMPLE

Two-Person Lift

I (6, 12)1/

/-I

(2, 4) /, I/

1/l O

O 1 2 3 4 5 6 7 X

Time (seconds)

1. WHAT IF? In Example 1, the cost is represented by y = 12x.Graph the equation and interpret the slope.

2. In Example 2, how much would a spacecraft that weighs3500 kilograms on Earth weigh on Titan?

The distance y (in meters) that a four-person ski lift travels in x seconds is represented by the equation y = 2.5x. The graph shows the distance that a two-person ski lift travels.

a. Which ski lift is faster?

Interpret each slope as a unit rate.Four-Person Lift

y = 2.5x t

( The slope is �

The four-person lift travels 2.5 meters per second.

Two-Person Lift

l change inys ope= change inx

=!!.=2 4

The two-person lift travels 2 meters per second.

::• So, the four-person lift is faster than the two-person lift.

b. Graph the equation that representsthe four-person lift in the samecoordinate plane as the two-personlift. Compare the steepness of thegraphs. What does this mean in thecontext of the problem?

::• The graph that represents thefour-person lift is steeper than the graph that represents the two-person lift. So, the four-person lift is faster.

Ski Lift

..-- y I '=i=!==:+� ':;:°::X'-,L. � 12 l-di 1 0 1------1-+---+·--¥--/ .s 8 l---1'---4----1�'--+

e 6 1----f--l+il"--'--",--f-........ _-l � 4 � 0

1 2 3 4 5 6 7 X

Time (seconds)

if On Your Own

�='t'.ou're9g, d a. Y Exercise 9

3. The table shows thedistance y (in meters) that a T-bar ski lift travels in x seconds. Compare its speed to the ski lifts in Example 3.

Section 4.3

x (seconds)

y (meters)

1

2_!_4

3 4

3 6- 9

4

Graphing Proportional Relationships 161

164

You can use a process diagram to show the steps involved in a procedure. Here is an

example of a process diagram for graphing a linear equation.

IGraphing a

]linear equation

IMake a table

)of values.

-1,

Plot the ordered

pairs.

,!,

Draw a line through the points.

Make process diagrams with examples to

help you study these topics.

1. finding the slope of a line

2. graphing a proportional relationship

After you complete this chapter, make

process diagrams for the following topics.

3. graphing a linear equation using

a. slope andy-intercept

b. x- and y-intercepts

4. writing equations in slope-intercept form

5. writing equations in point-slope form

�

Example Graphy=2x-1.

, y•(1.1)t

-3 -2 _, , 2 3 )(

LL -2 (0, -1)

(-1, -3)_

I I '

�ee hyena. �

$tancl very Still. ,&

$camper away!

"Here is a process dlogram with suggestions for what to do if a hyena

knocks on your door."


Graph the linear equation. (Section 4.1) ™81©!11� Progress Check '#'

BigideasMath com

1. y=-x+8X

2. y =- - 43

3. X = -1

Find the slope of the line. (Section 4.2)

5. r-+-�+ 6.

3

2

-3 -2 -1 I 2 3 X

7. 8.

(-2, 3)

-5 -4 -3 -1

( 2, -1)-2-

L3

6-

�2.-4) 5

3

2

1-

-3 -2 -1

4. y = 3.5

(3, 4)

l 2 3 X

9. What is the slope of a line that is parallel to the line in Exercise S?What is theslope of a line that is perpendicular to the line in Exercise 5? (Section 4.2)

10. Are the lines y = -1 and x = 1 parallel? Are they perpendicular? Justifyyour answer. (Section 4.2)

11. BANKING A bank charges $3 each time you use an out-of-network ATM.

At the beginning of the month, you have $1500 in your bank account. Youwithdraw $60 from your bank account each time you use an out-of-network ATM. Graph a linear equation that represents the balance in your accountafter you use an out-of-network ATM x times. (Section 4.1)

--

12. MUSIC The number y of hours of cello lessons that you takeafter x weeks is represented by the equation y = 3x. Graphthe equation and interpret the slope. (Section 4.3)

13. DINNER PARTY The cost y (in dollars) to provide food forguests at a dinner party is proportional to the number x

of guests attending the party. It costs $30 to provide foodfor 4 guests. (Section 4.3)

a. Write an equation that represents the situation.

b. Interpret the slope.

c. How much does it cost to provide food for 10 guests?

Sections 4.1-4.3 Quiz 165

Key Vocabulary .. x-intercept, p. 168

y-intercept, p. 168

slope-intercept form,p. 168

Linear equations can, but do not alw ays, pass through the origin. So, p roportional relationships are a special type of linear equation in which b = 0.

EXAMPLE

'J Key Ideas Intercepts

The .x.intercept of a line is the x-coordinate of the point wherethe line crosses the x-axis. It occurswheny= 0.

.Y�---y Intercept - b

x-intercept = a

They-intercept of a line is the y-coordinate of the point wherethe line crosses the y-axis. It occurs whenx = 0.

Slope-Intercept Form

0

Words A linear equation written in the form y = mx + bis in slope-intercept form. The slope of the line is m,

and they-intercept of the line is b.

Algebra y = mx + lJ.. ..

(slop�) y-intercept

X

Find the slope and they-intercept of the graph of each linear equation.

a. y = -4x- 2

y = -4x + ( 2) Write in slope-intercept form.

::• The slope is -4, and they-intercept is -2.

3 b. y- 5 = -x2 3 y= x+ 5 2

Add 5 to each side.

::• The slope is�, and they-intercept is 5. 2

(ii, On Your Own

Find the slope and they-intercept of the graph of the linear equation.

2 1. y = 3x - 7 2. y - l = -

3x

168 Chapter 4 Graphing and Writing Linear Equations ◄ MuHi•Language Glossary at BigideasMathifcom

EXAMPLE

4

Graph y = -3x + 3. Identify the x-intercept.

Step 1: Find the slope and they-intercept.

y = -3x + :�

� �

\ y - -3x - 3]

Step 2: They-intercept is 3. So, plot (0, 3).

Step 3: Use the slope to find another point and draw the line.

\7 \.

-3 .....__._ ____ \ ___ _, \

3

rise -3m=-=-

run l

Plot the point that is l unit right -3-2-1

3

3 X

\ -2

and 3 units down from (0, 3). Draw ....--.--,..--2•-.....__._...__..___,

EXAMPLE

a line through the two points.

::• The line crosses the x-axis at (1, 0). So, the x-intercept is 1.

The cost y (in dollars) of taking a taxi x miles is y = 2.5x + 2.(a) Graph the equation. (b) Interpret they-intercept and the slope.

a. The slope of the line is 2.5 =%-Use the slope and they-interceptto graph the equation.

y -

-$EXTRA - --- $ FARE --- 7•-

.---; r· r,C. .://_/ I They-intercept is 2. 6 ['..,

I/ -r---- Use the slope to plot RECORD �

-- Dollars ---C4l""ts -So, plot (0, 2).

�

another point, (2, 7). Draw I

Time Rate Extra

* ◄ ,,.

� Y.ou;re Rta.dyExercises 18-2 3

I a line through the points.

(0, 2) -2 -�

0 0 l 2 3 4 5 6 x

b. The slope is 2.5. So, the cost per mile is $2.50. They-interceptis 2. So, there is an initial fee of $2 to take the taxi.

� On Yoqr Own.

Graph the linear equation. Identify the x-intercept. Use a graphing

calculator to check your answer.

1 3. y = x - 4 4. y = --x, l

2

5. In Example 3, the cost y (in dollars) of taking a different taxix miles is y = 2x + 1.5. Interpret they-intercept and the slope.

Section 4.4 Graphing Linear Equations in Slope-Intercept Form 169

Key Vocabulary '4 standard form, p. 174

Any linear equation can be written in standard form.

EXAMPLE

Cheek

.,,.---------

.,.,.---_.,..- ��---�

. ...----- -2x-t-3y=-6

-6

r) l<ey IdeaStandard Form of a Linear Equation

The standard form of a linear equation is

ax+by=c

where a and b are not both zero.

Graph-2x + 3y = -6. Step l: Write the equation in slope-intercept form.

-2x + 3y = -6

3y = 2x - 6

2 y =

3x- 2

Write the equation.

Add 2x to each side.

Divide each side by 3.

Step 2: Use the slope and they-intercept to graph the equation.

2 y = X + (-2)

4

3 A (§--l Y y-intercept)

y ,-.._,-.--,-,... 2 -r-r--t-

.._L)

They-intercept is 2. So, plot (0, -2)

Use the slope to plot another point, (3, O).


-» On Your Own

174

Now...Y.OLl'ro �?ad --�y

Exercises 5-1 O

Graph the linear equation. Use a graphing calculator to check your graph.

1. X + y= -2 2. 1

--x + 2y = 62

3. 2

--x + y= 0 4. 2x+y=5

Chapter 4 Graphing and Writing Linear Equations ◄ Multi-Lan9ua9e Glossary at BigideasMath�m

EXAMPLE

Check 2

Graph x + 3y = -3 using intercepts.

Step 1: To find the x-intercept, substitute O for y.

X + 3y = -3

X + 3(0) = -3

x= -3

Step 2: Graph the equation.

To find they-intercept, substitute O for x.

X + 3y = -3

0 + 3y = -3

y= -1

They-intercept is 1. So, plot (O, -1 ).

-•~� .. • 4-------..

x---3y=-3

The x-intercept is

-3. So, plot ( 3, O).-3

-6

EXAMPLE ID

E>ananas

SO.GO/pound

I Apples

$1.50/pound

+---,,--+--+--l 4 �,-.....j.--------+-"-,-l

ls l.--l--'---1-__...---i..

1_6 Draw a line through the points.

You have $6 to spend on apples and bananas. (a} Graph the equation l .5x + 0.6y = 6, where xis the number of pounds of apples and y is thenumber of pounds of bananas. (b} Interpret the intercepts.

a. Find the intercepts and graph the equation.

x-intercept

l.5x + 0.6y = 6l.5x + 0.6(0) = 6

x=4

y-intercept

l.5x + 0.6y = 6 1.5(0) + 0.6y = 6

y= 10

)' 12· (0, 1 O)flO , JJII_,, : · 1.Sx-0.6y=6

: 1--+-�-l>.---� J 2· . (4, O)

l 2 3 4 5 6 X

b. The x-intercept shows that you can buy 4 pounds of apples whenyou do not buy any bananas. They-intercept shows that you canbuy 10 pounds of bananas when you do not buy any apples.

Wf On Your Own

�Qcly Exercises 16-18

Graph the linear equation using intercepts. Use a graphing calculator to check your graph.

5. 2X - y = 8 6. X + 3y = 6

7. WHAT IF? In Example 3, you buy y pounds of oranges instead of bananas. Oranges cost $1.20 per pound. Graph the equation l.Sx + l.2y = 6. Interpret the intercepts.

Section 4.5 Graphing Linear Equations in Standard Form 175

EXAMPLE <[]

After writing an equation, check that the given points are solutions of the equation.

™·� Lesson Tutorials '¥ BigideasMath com

Write an equation of the line in slope-intercept form.

a.

b.

Find the slope and they-intercept.

J\ Y1 m=--

x2 -xi 2-52-0

3 3 =- or--2 ' 2

Because the line crosses the y-axis at (0, 5), they-intercept is 5. � r -�

y-intercept

' S h . . 3 � ::• o, t e equat10n1sy= -2x+ ,).

Find the slope and they-intercept. V - V

m = :....L_:_!_

x2 - x 1

-3-2

0-3

- 5 5 =- or-3, 3

Because the line crosses the y-axis at (0, -3), they-intercept is -3.

�

::• So, the equation is y = ¾x + (

y-intercept

3),ory=�x-3.3

(i On Your Own

�dy Exercises 5-1 O


1. 2.

180 Chapter 4 Graphing and Writing Linear Equations

EXAMPLE �

The graph of y = a isa horizontal line that passes through (O, a).

EXAMPLE

Engineers used tunnel boring machines like the ones shown above to dig an extension of the Metro Gold Line in Los Angeles. The new tunnels are 1 .7 miles long and 21 feet wide.

Which equation is shown in the graph?

@ y= -4 @ y= -3 © y=O @ y= -3x

Find the slope and they-intercept.

The line is horizontal, so the change in y is 0.

m = change in y = Q = 0change in x 3

-4 -3 -2 -1 1 2 X ,__,--+--1+--r-+-

t----',---+--+- I_ z-1-......_+--

( 3, 4)--3 (0, 4)

,-..;,---+--+- - 51--+-----

Because the line crosses the y-axis at (0, -4), they-intercept is -4.

::• So, the equation is y = Ox + ( 4), or y = -4. The correct answer is @.

The graph shows the distance remaining to complete a tunnel. (a) Write an equation that representsthe distance y (in feet) remaining

after x months. (b) How much time

does it take to complete the tunnel?

a. Find the slope and they-intercept.

m = change lny = -2000 = _500change in x 4

Because the line crosses the y-axis

at (0, 3500), they-intercept is 3500.

..., )'

Tunnel Digging

] 4000 (0 3 500). ...... 3 soo�'+--+--'----+--'--,f-----,

-g 3000'iii 25001--f--.->l..,_. __ �--+---IE 20001--f---,f---'l.--1..,......---..-+-� 1500

� 1000 t; S001--1---J,____.__,--+--'l,--+-

OO 1 2 3 4 5 6 7 X

Time (months)

::• So, the equation is y = -500x + 3500.

b. The tunnel is complete when the distance remaining is O feet.

So, find the value of x when y = 0.

y = -500x + 3500

O = -500x + 3500

-3500 = -500x

7=x

Write the equation.

Substitute 0 for y.

Subtract 3500 from each side.

Divide each side by - 500.

::• It takes 7 months to complete the tunnel.

t, On You.r Own

ito�-Yo.u'r,,:, t>- d \�y Exercises 13-15 3. Write an equation of the line that passes through (0, 5) and (4, 5).

4. WHAT IF? In Example 3, the points are (0, 3500) and (5, 1500).

How long does it take to complete the tunnel?

Section 4.6 Writing Equations in Slope-Intercept Form 181

Key Vocabulary '4

point-slope form, p. 186

EXAMPLE <[)

'l<ey Idea Point-Slope Form

Words A linear equation written in the form y - y1 = m(x - x1)

is in point-slope form. The line passes through the point(x

1, y

1 ), and the slope of the line ism.

Algebra y - Y1 = lll(X - -') A A

,,.- � passes through (x1, y1)

)'

y- Y1

X

Write in point-slope form an equation of the line that passes through the point (-6, 1) with slope�-

y - I, = 1ll(X - .\)

)

y - 1 = - lx - ( 6) l .I

2 y- 1 = -(x + 6) 3

Write the point-slope form

Substitute ! form, 6 for x1, and 1 for y

1.

Simplify.

::• So, the equation is y- 1 = �(x + 6). 3

Check Check that (-6, 1) is a solution of the equation.

2 y- 1 = -(X + 6) 3

1 - 1 == �( G + 6)3

O=O ✓

Write the equation.

Substitute.

Simplify.

4i, On Your Own

186

!!I.ow JXl�cJ . yExercises 6-11

Write in point-slope form an equation of the line that passes through the given point and has the given slope.

1. (1, 2); m = -4 2. (7, 0); m = 1 3 3. (-8, -5); m = --4


EXAMPLE �

You can use either of the given points to write the equation of the line.

Use m = -2 and(5, -2).

y - (-2) = -2(x - 5)

y+2=-2x+10

y- -2x + 8 ✓

EXAMPLE ID

10 feet per

second

Write in slope-intercept form an equation of the line that passes

through the points (2, 4) and (5, -2).

Find the slope: m = _Y_.l_, =2 4

= � = -2X2 X

1 5 - 2 3

Then use the slope m = -2 and the point (2, 4) to write an equation of the line.

Y - l' = m (x - \ )• I

Write the point-slope form.

y-4= 2(x- 2)

y- 4 = -2x + 4

y= -2x- 8

Substitute 2 form, 2 for x1, and 4 for r,.

Distributive Property

Write in slope-intercept form.

You finish parasaiJing and are being pulled back to the boat. After

2 seconds, you are 25 feet above the boat. (a) Write and graph an equation that represents your heighty (in feet) above the boat after

x seconds. (b) At what height were you parasailing'?

a. You are being pulled down at the rate of 10 feet per second.So, the slope is -10. You are 25 feet above the boat after 2 seconds.So, the line passes through (2, 25). Use the point-slope form.

y-25= l0(x- 2)

y - 25 = - lOx + 20

y = -lOx + 45

Substitute form, x1, and y

1.


Write in slope-intercept form.

::• So, the equation is y = -lOx + 45.

b. You start descending when x = 0. They-intercept is 45. So, you were parasailingat a height of 45 feet.

y

45 ,ox - 45

40

351.--�'-+--+--+--+--+--l

30

25

20

151-'--t-\c--+----+--+--..--l

10

5

t, On Your OwnO

O I 2 3 4 5 6 7 X

Write in slope-intercept form an equation of the line that passes

through the given points.

4. (-2,1),(3,-4) 5. (-5,-5),(-3,3) 6. (-8,6).(-2,9)

7. WHAT IF? In Example 3, you are 35 feet above the boat after2 seconds. Write and graph an equation that represents yourheight y (in feet) above the boat after x seconds.

Section 4.7 Writing Equations In Point-Slope Form 187

190

'

Progress Check

Big!deasMath Find the slope and they-intercept of the graph of the linear equation. (Sect io11 .:J.-4)

11. y=-x-8 2. y=-x+3

4

Find the x- and y-intercepts of the graph of the equation. (!:;ection .J.5)

3. 3X - 2y = 12 4. X + 5y = 15

Write an equation of the line in slope-intercept form. (Sectio11 4.6)

5.

.J�

J

'

6 f-s .\' '�I 7. � )'

5 -4 -3

-3

2 -r---

--------3

,-2

4

5 X

- -5

Write in point-slope form an equation of the line that passes through the given point and has the given slope. 1�ec1w11 71

8. (1,3); m = 21 9. (-3, -2); m = -3

10. (-1,4); rn = -11

11. (8, -5); m = -8

Write in slope-intercept form an equation of the line that passes through the given points. (Section 4. 1)

12. (0 -�) (-3 -�)' 3 ' 3 13. (4, 0), (0, 4)

14. STATE FAIR The cost y (in dollars) of one person buyingadmission to a fair and going on x rides is y = x + 12.rSe<tio11 14)

a. Graph the equation.

b. Interpret they-intercept and the slope.

15. PAINTING You used $90 worth of paint for a school float.(�ectiOII 4 5)

a. Graph the equation 18x - 15y = 90, where xis thenumber of gallons of blue paint and y is the numberof gallons of white paint.

b. Interpret the intercepts.

-�

�

16. CONSTRUCTION A construction crew is extending a highway sound barrier

that is 13 miles long. The crew builds .! of a mile per week. Write an equation2

that represents lhe lengthy (in miles) of the barrier after x weeks. (5ecr1m1 4.6)


•

Review Key Vocabulary

linear equation p. 144

solution of a linear equation. p. 144

slope, p. 150

rise, p. 7 50

run, p. 150

Review Examples and Exercises

' Vocabulary Help

BigideasMath

x-intercept, p. 168

y-intercept, p. 168

slope-intercept form, p. 168

standard form, p. 174

point-slope form, p. 186

cU:, i) Graphing Linear Equations (pp. 742-747)

Graphy = 3x - I.

Step 1: Make a table of values.

X y = 3x - 1 y (x, y)

2 y=3(-2)-l -7 (-2, -7)

-1 y=3(-l)-l 1 (-1, -4)

0 y = 3(0) - 1 1 (0, -1)

1 y = 3(1) - 1 2 (1, 2)

Step 2: Plot the ordered pairs. y 2--.--

Step 3: Draw a line through the points.

11

f¥L

-5-4-3-2-1 I 2 3 4 x

I 1 (O, -1) 1� -2 r-+-

� - I---- -3+--.--+---i--------1 �-,d.-.___ ---+- -5

...__... _l_-6-,-+--t-+----, [ (-2, -7)1 -7 1

Exercises

Graph the linear equation.

3 1. y = -x

5

4. y= 1

2. y = -2

2 5. y= -x + 23

3. y= 9 - X

6. X = -5

Chapter Review 191

192

� Slope of a Line (pp. 148-157)

Find the slope of each line in the graph. 1 •

Red Lfr,e: m = · - "' = - ' · X2 - X1 2 - 2 0

:: The slope of the red line is undefined.

l' - I' I - 2 -3 3 Blue Line: m = 2 • 1 = --- = -, or --

x -x1

4-(-3) i 7

y-y 4 4 0 Green Line: m = -2--

1 = -- = or 0 - r: 0 _,

Exercises

_\2 •• \ I :., - 3

The points in the table lie on a line. Find the slope of the line.

]. I : I �I i � I � I : I a. I : I �2

! : i : I : I9. Are the lines x = 2 andy = 4 parallel? Are they perpendicular? Explain.

�U3 Graphing Proportional Relationships (pp. 158-163)

The cost y (in dollars) for x tickets to a movie is represented by the equation

y = 7x. Graph the equation and interpret the slope.

The equation shows that the slope mis 7. So, the graph passes through (0, 0) and (1, 7).

Plot the points and draw a line through the points. Because negative values of x do not make sense in this context, graph in the first quadrant only .

.

::• The slope indicates that the unit costis $7 per ticket.

Exercises

Movie Tickets

0 1 2 3 4 5 6 X

Number of tickets

10. RUNNING The number y of miles you run after x weeks is represented bythe equation y = 8x. Graph the equation and interpret the slope.

11. STUDYING The number y of hours that you study after x days is representedby the equation y = l .5x. Graph the equation and interpret the slope.


� Graphing Linear Equations in Slope-Intercept Form (pp. 166-171)

Graph y = 0.5x - 3. Identify the x-intercept.

Step 1: Find the slope and they-intercept.

y = 0.5x + ( 3)

�peJ,_ ___ ___.t L_� y-intercept )

Step 2: They-intercept is -3. So, plot (0, -3).

Step 3: Use the slope to find another point and draw the line. _ rise _ 1

m----

run 2

Plot the point that is 2 units right and 1 unit up from (0, -3). Draw a line through the two points.

::• The line crosses the x-axis at (6, 0). So, the x-intercept is 6.

Exercises

Graph the linear equation. Identify the x-intercept. Use a graphing calculator to check your answer.

12. y=2x-6 13. y = -4x + 8 14. y= -x- 8

a� Graphing Linear Equations in Standard Form (pp. 112-177)

Graph 8x + 4y = 16.

Step 1: Write the equation in slope-intercept form.

8x + 4y = 16 4y = -8x + 16 y = -2x + 4

Write the equation . Subtract Bx from each side.Divide each side by 4.

Step 2: Use the slope and they-intercept to graph the equation.

y=-2x+4

( slope )t----� �

They-intercept is 4.So, plot (O, 4).


Use the slope to plotanother point, (1, 2).

Chapter Review 193

194

Exercises

Graph the linear equation.

1 15. -x+y=3

416. -4x + 2y = 8

17. XT5y=l01 l 3

18. --x+ y=-2 8 4

19. A dog kennel charges $30 per night to board your dog and $6 for each hourof playtime. The amount of money you spend is given by 30x + 6y = 180,

where xis the number of nights and y is the number of hours of playtime.Graph the equation and interpret the intercepts.

a@ Writing Equations in Slope-Intercept Form (pp 118-183)


a. y Find the slope and they-intercept. �6

P=-..!(2,4

) �2 17

� 1 2 3 I .1. I

Y� Yi -1 2 2 m=--=--=- orl · 2 0 ?'.\2 .\ I - �

Because the line crosses the y-axis at (0, 2), they-intercept is 2.

� ,... y-mtercept

,. ,.

::• So, the equation is y = lx + 2, or y = x + 2.

b. --i I Find the slope and they-intercept.

Chapter 4

l' - - I ( 2) .. 2 m= ·

2 =----=- or--·t':i t, 3 0 3 ' 3

Because the line crosses they-axis at (0, -2), they-intercept is -2.

·· s th · · 2

2) 2

2 :•• o, e equation 1s y = -3x + (- , or y = -3x - .

Graphing and Writing Linear Equations

Exercises


20. 21.

22.

-2-1_1 �2 3 4 X

-2(2, 3)

-3 :7

23.

---rl

-1 Ll.._3 4 5 X

-2 -I - I

h : 2.i-,......--.��_,

a 3..i...:(....:..O•--.:�.J..__J

24. Write an equation of the line that passes through (0, 8) and (6, 8).

25. Write an equation of the line that passes through (0, -5) and (-5, -5).

fiJot} Writing Equations in Point-Slope Form (pp 184-189)

Write in slope-intercept form an equation of the line that passes through the points (2, 1) and (3, 5).

Find the slope. l' -

5 l 4m = ·

2 = -- = - or 4

3 2 1, X2 -XI

Then use the slope and one of the given points to write an equation of the line.

Use m = 4 and (2, 1).

Write the point-slope form.

6

....5

4

-I

)'

y - y1

= 111(x - x)

y- 1 = -l(x - 2)

y- 1 = 4x - 8

Substitute 4 form, 2 for x1, and 1 for y

1.


y = 4x - 7 Write in slope-intercept form.

::• So, the equation is y = 4x - 7.

Exercises

26. Write in point-slope form an equation of the line that passesthrough the point (4, 4) with slope 3.

27. Write in slope-intercept form an equation of the line that passesthrough the points (-4, 2) and (6, -3).

5 X

Chapter Review 195

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