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Name   Class    Date    3-6Lines in the Coordinate PlaneGoing DeeperEssential question: How can you use slope to write equations of lines that are parallel or perpendicular?

Recall that a linear function can be expressed as a linear equation. You can write a linear equation in different forms depending upon the information you are given and the problem you are trying to solve.

Writing Equations of Parallel Lines

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

A The line parallel to y = -2x + 3 that passes through (1, -4)

The given line is in slope-intercept form and its slope is .

The required line has slope because parallel lines have the same slope.

y - y 1 = m(x - x 1 ) Use point-slope form.

y - = (x - ) Substitute for m, x 1 , and y 1 .

y + = Simplify each side of the equation.

y = Write the equation in slope-intercept form.

B The line that passes through (2, 3) and is parallel to the line through (1, -2) and (7, 1)

The slope of the line through (1, -2) and (7, 1) is

m = y 2 - y 1

_____ x 2 - x 1 = -

_________ -

= ____ = ____ .

So, the required line has slope .

y - y 1 = m(x - x 1 ) Use point-slope form.

y - = (x - ) Substitute for m, x 1 , and y 1 .

y = Simplify and write slope-intercept form.

E X A M P L E1

          Slope-Intercept Form

The equation of a line with slope m and y-intercept b is y = mx + b.

          Point-Slope Form

The equation of a line with slope m that passes through the point ( x 1 , y 1 ) is y - y 1 = m(x - x 1 ).

G-GPE.2.5

Chapter 3 117 Lesson 6

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REFLECT

1a. In Part A, how can you check that you wrote the correct equation?

1b. In Part A, once you know the slope of the required line, how can you finish solving the problem using the slope-intercept form of a linear equation?

WritingEquationsofPerpendicularLines

Write the equation of the line perpendicular to y = 3x - 8 that passes through (3, 1). Write the equation in slope-intercept form.

A First find the slope of the required line.

The given line is in slope-intercept form and its slope is .

Let the required line have slope m. Since the lines are perpendicular, the product of their slopes is -1.

So, · m = -1, and therefore, m = .

B Now use point-slope form to find the equation of the required line.

y - y 1 = m(x - x 1 ) Use point-slope form.

y - = (x - ) Substitute for m, x 1 , and y 1 .

y - = Distributive Property

y = Write the equation in slope-intercept form.

REFLECT

2a. How do you find the slope of the given line?

2b. How can you use graphing to check your answer?

E X A M P L E2G-GPE.2.5

Chapter 3 118 Lesson 6

y8

6

4

2

0

x

-8 -6 -4 -2

-8

-6

-4

-28642

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2c. Confirm your answer by graphing on the grid below.

p r a c t i c eWrite the equation of each line in slope-intercept form.

5. The line parallel to y = 5x + 1 that passes through (3, 8)

7. The line that passes through (-1, 0) and is parallel to the line through (0, 1) and (2, -3)

9. The line parallel to x - 3y = -12 that passes through (-3, 4)

6. The line parallel to y = -3x -2 that passes through (-2, 7)

8. The line that passes through (3, 5) and is parallel to the line through (3, 3) and (-3, -1)

10. The line parallel to 3x + y = 8 that passes through (0, -4)

1. The line with slope 3 that passesthrough (0, 6)

3. The line with slope -1 that passes through (3, 5)

2. The line with slope -4 that passesthrough (0, -5)

4. The line with slope 5 that passes through (2, -5)

11. Use the slope-intercept form of a linear equation to prove that if two lines are parallel then they have the same slope. (Hint: Use an indirect proof. Assume the lines have different slopes, m 1 and m 2 . Write the equations of the lines and show that there must be a point of intersection.)

Chapter 3 119 Lesson 6

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18. ErrorAnalysis Astudentwasaskedtofindthe equationofthelineperpendiculartoy-2x=1thatpassesthroughthepoint(4,3).Thestudent’sworkisshownatright.Explaintheerrorandgivethecorrectequation.

19. Arethelinesgivenbytheequations-4x+y=5and-x+4y=12parallel,perpendicular,orneither?Why?

20. ConsiderthepointsA(-7,10),B(12,7),C(10,-24),andD(-8,-3).Whichtwolinesdeterminedbythesepointsareperpendicular?Explain.

16. Thelineperpendicularto2y=x+5thatpassesthrough(2,1)

17. Thelineperpendicularto3x+y=8thatpassesthrough(0,-2)

Thegivenlinehasslope-2,sotherequiredlinehasslope1__2.

y-y1=m(x-x1) Use point-slope form.

y-3= 1 __ 2 (x-4) Substituteform,x1,y1.

y-3= 1 __ 2 x-2 Distributive Property

y= 1 __ 2 x+1 Add 3 to both sides.

12. Thelineperpendiculartoy=1_2x+1thatpassesthrough(1,4)

14. Thelinethatpassesthrough(1,2)andisperpendiculartothelinethrough(3,-2)and (-3,0)

13. Thelineperpendiculartoy=-x+2thatpassesthrough(-1,-7)

15. Thelinethatpassesthrough(-2,3)andisperpendiculartothelinethrough(0,1)and(-3,-1)

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

Chapter 3 120 Lesson 6

Name ________________________________________ Date __________________ Class__________________

Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.

Holt McDougal Geometry

Practice Lines in the Coordinate Plane

Write the equation of each line in the given form.

1. the horizontal line through (3, 7) in 2. the line with slope 85

− through (1, −5) in

point-slope form point-slope form

_________________________________________ _________________________________________

3. the line through 1 7,2 2

− − and (2, 14) in 4. the line with x-intercept −2 and y-intercept

slope-intercept form −1 in slope-intercept form

_________________________________________ _________________________________________

Graph each line.

5. 33 ( 1)4

y x+ = + 6. 4 23

y x= − +

Determine whether the lines are parallel, intersect, or coincide.

7. 15 0, 1 ( 5)5

x y y x− = + = + ____________________

8. 12 2 , 12

y x x y+ = = − + ____________________

9. 3 14( 3), 44 4

y x y x= − + = − ____________________

An aquifer is an underground storehouse of water. The water is in tiny crevices and pockets in the rock or sand, but because aquifers underlay large areas of land, the amount of water in an aquifer can be vast. Wells and springs draw water from aquifers. 10. Two relatively small aquifers are the Rush Springs (RS) aquifer and the Arbuckle-

Simpson (AS) aquifer, both in Oklahoma. Suppose that starting on a certain day in 1985, 52 million gallons of water per day were taken from the RS aquifer, and 8 million gallons of water per day were taken from the AS aquifer. If the RS aquifer began with 4500 million gallons of water and the AS aquifer began with 3000 million gallons of water and no rain fell, write a slope-intercept equation for each aquifer and find how many days passed until both aquifers held the same amount of water. (Round to the nearest day.)

_________________________________________________________________________________________

20

LESSON

3-6

CS10_G_MEPS710006_C03PWBL06.indd 20 4/21/11 5:58:59 PM

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3-6Name   Class    Date   

Additional Practice

Chapter 3 121 Lesson 6

Name ________________________________________ Date __________________ Class__________________

Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.

Holt McDougal Geometry

Problem Solving Lines in the Coordinate Plane

Use the following information for Exercises 1 and 2. Josh can order 1 color ink cartridge and 2 black ink cartridges for his printer for $78. He can also order 1 color ink cartridge and 1 black ink cartridge for $53. 1. Let x equal the cost of a color ink 2. What is the cost of each cartridge?

cartridge and y equal the cost of a ________________________________ black ink cartridge. Write a system of equations to represent this situation.

_________________________________________

3. Ms. Williams is planning to buy T-shirts for the cheerleading camp that she is running. Both companies’ total costs would be the same after buying how many T-shirts? Use a graph to find your solution.

Art Creation Fee

Cost per T-shirt

Company A $70 $10

Company B $50 $12

_________________________________________

_________________________________________

Choose the best answer. 4. Two floats begin a parade at different 5. A piano teacher charges $20 for each half

times, but travel at the same speeds. hour lesson, plus an initial fee of $50. Which is a true statement about the Another teacher charges $40 per hour, lines that represent the distance traveled plus a fee of $50. Which is a true statement by each float at a given time? about the lines that represent the total cost by each piano teacher?

6. Serina is trying to decide between two similar packages for starting her own Web site. Which is a true statement? A Both packages cost $235.50 for 5 months. B Both packages cost $295 for 10 months. C Both packages cost $355 for 15 months. D The packages will never have the same cost.

F The lines intersect. G The lines are parallel. H The lines are the same. J The lines have a negative slope.

Design and Setup

Monthly Fee to Host

Package A $150.00 $14.50

Package B $175.00 $12.00

A The lines intersect. B The lines are parallel. C The lines are the same. D The lines have a negative slope.

104

LESSON

3-6

CS10_G_MEPS710006_C03PSL06.indd 104 5/19/11 11:02:30 PM

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Problem Solving

Chapter 3 122 Lesson 6