Holt Geometry 3-6 Lines in the Coordinate Plane 3-6 Lines in the Coordinate Plane Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation

Holt Geometry

3-6 Lines in the Coordinate Plane3-6 Lines in the Coordinate Plane

Holt Geometry

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Geometry

3-6 Lines in the Coordinate Plane

Warm UpSubstitute the given values of m, x, and y into the equation y = mx + b and solve for b.

1. m = 2, x = 3, and y = 0

Solve each equation for y.

3. y – 6x = 9

2. m = –1, x = 5, and y = –4

b = –6

b = 1

4. 4x – 2y = 8

y = 6x + 9

y = 2x – 4

Holt Geometry


Graph lines and write their equations in slope-intercept and point-slope form.

Classify lines as parallel, intersecting, or coinciding.

Objectives

Holt Geometry


point-slope formy –y1 = m(x – x1)

slope-intercept formy = mx + b

Vocabulary

Holt Geometry


The equation of a line can be written in many different forms. The point-slope and slope-intercept forms of a line are equivalent. Because the slope of a vertical line is undefined, these forms cannot be used to write the equation of a vertical line.

Holt Geometry


Holt Geometry


A line with y-intercept b contains the point (0, b).A line with x-intercept a contains the point (a, 0).

Remember!

Holt Geometry


A system of two linear equations in two variables represents two lines. The lines can be parallel, intersecting, or coinciding. Lines that coincide are the same line, but the equations may be written in different forms.

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Holt Geometry


Example 1A: Writing Equations In Lines

Write the equation of each line in the given form:

a line with slope 6 through (3, –4) inpoint-slope form

y – y1 = m(x – x1)

y – (–4) = 6(x – 3)

Point-slope form

Substitute 6 for m, 3 for x1, and -4 for y1.

Holt Geometry


Example 1B: Writing Equations In Lines


a line through (–1, 0) and (1, 2) inslope-intercept form

y = mx + b

0 = 1(-1) + b

1 = b

y = x + 1

Slope-intercept form

Find the slope.

Substitute 1 for m, -1 for x, and 0 for y.

Write in slope-intercept form using m = 1 and b = 1.

2 1

2 1

y ym

x x

Holt Geometry


Example 1C: Writing Equations In Lines


a line with x-intercept 3 and y-intercept –5in point slope form

y – y1 = m(x – x1) Point-slope form

Use the points (3,0) and (-5,0) to find the slope.

Simplify.

Substitute for m, 3 for x1, and 0 for y1.

53

y = (x - 3)53

y – 0 = (x – 3)53

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Check It Out! Example 1D


a line with slope 0 through (4, 6)in slope-intercept form

y = 6

y – y1 = m(x – x1)

y – 6 = 0(x – 4)

Point-slope form

Substitute 0 for m, 4 for x1, and 6 for y1.

Holt Geometry


Graph each line.

Example 2A: Graphing Lines

The equation is given in the

slope-intercept form, with a

slope of and a y-intercept of

1. Plot the point (0, 1) and

then rise 1 and run 2 to find

another point. Draw the line

containing the points.

(0, 1)

rise 1

run 2

Holt Geometry


Check It Out! Example 2B

Graph each line.

y = 2x – 3


slope-intercept form, with a

slope of and a y-intercept of

–3. Plot the point (0, –3) and

then rise 2 and run 1 to find

another point. Draw the line

containing the points.

(0, –3)

rise 2

run 1

Holt Geometry


Graph each line.

Example 2C: Graphing Lines

y – 3 = –2(x + 4)


point-slope form, with a slope of

through the point (–4, 3). Plot

the point (–4, 3) and then rise –

2 and run 1 to find another

point. Draw the line containing

the points.

(–4, 3)

rise –2

run 1

Holt Geometry


Graph each line.

Example 2D: Graphing Lines

The equation is given in the form

of a horizontal line with a

y-intercept of –3.

The equation tells you that the

y-coordinate of every point on the

line is –3. Draw the horizontal line

through (0, –3).

y = –3

(0, –3)

Holt Geometry


Determine whether the lines areparallel, intersect, or coincide.

Example 3A: Classifying Pairs of Lines

y = 3x + 7, y = –3x – 4

The lines have different slopes, so they intersect.

Holt Geometry


Example 3B: Classifying Pairs of Lines

Solve the second equation for yto find the slope-intercept form.

6y = –2x + 12

Both lines have a slope of , and the y-intercepts are

different. So the lines are parallel.


Holt Geometry


Example 3C: Classifying Pairs of Lines

2y – 4x = 16, y – 10 = 2(x - 1)

Solve both equations for yto find the slope-intercept form.

2y – 4x = 16

Both lines have a slope of 2 and a y-intercept of 8, so they coincide.

2y = 4x + 16

y = 2x + 8

y – 10 = 2(x – 1)

y – 10 = 2x - 2y = 2x + 8


Holt Geometry


Check It Out! Example 3D

Determine whether the lines 3x + 5y = 2 and3x + 6 = -5y are parallel, intersect, or coincide.

Both lines have the same slopes but different y-intercepts, so the lines are parallel.

Solve both equations for y tofind the slope-intercept form.

3x + 5y = 2

5y = –3x + 2

3x + 6 = –5y

Holt Geometry


Erica is trying to decide between two car rental plans. For how many miles will the plans cost the same?

Example 4: Problem-Solving Application

/mi

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11 Understand the Problem

The answer is the number of miles for which the costs of the two plans would be the same. Plan A costs $100.00 for the initial fee and $0.35 per mile. Plan B costs $85.00 for the initial fee and $0.50 per mile.

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22 Make a Plan

Write an equation for each plan, and then graph the equations. The solution is the intersection of the two lines. Find the intersection by solving the system of equations.

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Solve33

Plan A: y = 0.35x + 100

Plan B: y = 0.50x + 85

0 = –0.15x + 15

x = 100

y = 0.50(100) + 85 = 135

Subtract the second equation from the first.

Solve for x.

Substitute 100 for x in the first equation.

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The lines cross at (100, 135).

Both plans cost $135 for 100 miles.

Solve Continued33

Holt Geometry


Check your answer for each plan in the original problem.

For 100 miles, Plan A costs $100.00 + $0.35(100) = $100 + $35 = $135.00.

Plan B costs $85.00 + $0.50(100) = $85 + $50 = $135, so the plans cost the same.

Look Back44

Holt Geometry


Check It Out! Example 4

What if…? Suppose the rate for Plan B was also $35 per month. What would be true about the lines that represent the cost of each plan?

The lines would be parallel.

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Lesson Quiz: Part I

Write the equation of each line in the given form. Then graph each line.

1. the line through (-1, 3) and (3, -5) in slope-intercept form.y = –2x + 1

25

y + 1 = (x – 5)

2. the line through (5, –1) with slope in point-slope form.

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Lesson Quiz: Part II

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

3. y – 3 = – x,

intersect

12 y – 5 = 2(x + 3)

4. 2y = 4x + 12, 4x – 2y = 8

parallel

Documents

Holt Geometry 3-6 Lines in the Coordinate Plane 3-6 Lines in the Coordinate Plane Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation