Lines in the Coordinate PlaneLines in the Coordinate Plane

Holt McDougal Geometry

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

Holt Geometry

Warm Up

Lesson Presentation

Lesson Quiz



3-6 Lines in the Coordinate Plane

Warm Up Substitute 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



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

Classify lines as parallel, intersecting, or coinciding.

Objectives



point-slope form

slope-intercept form

Vocabulary



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.



4, 3



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

Remember!



Example 1A: Writing Equations In Lines

Write the equation of each line in the given form.

the line with slope 6 through (3, –4) in point-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.



Example 1B: Writing Equations In Lines


the line through (–1, 0) and (1, 2) in slope-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.



Example 1C: Writing Equations In Lines


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

y – y1 = m(x – x1) Point-slope form

Use the point (3,-5) to find the slope.

Simplify.

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

5

3

y = (x - 3) 5

3

y – 0 = (x – 3) 5

3



Check It Out! Example 1a


the 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.



Check It Out! Example 1b


the line through (–3, 2) and (1, 2) in point-slope form

y - 2 = 0

Find the slope.

y – y1 = m(x – x1) Point-slope form

Simplify.

Substitute 0 for m, 1 for x1, and 2 for y1.

y – 2 = 0(x – 1)



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



Graph each line.

Example 2B: 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



(–4, 3)

rise –2

run 1



Graph each line.

Example 2C: 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)



Check It Out! Example 2a

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



Check It Out! Example 2b

Graph each line.


point-slope form, with a slope

of through the point (–2, 1).

Plot the point (–2, 1)and then

rise –2 and run 3 to find



(–2, 1)

run 3

rise –2



Check It Out! Example 2c

Graph each line.

y = –4

The equation is given in the form

of a horizontal line with a

y-intercept of –4.

The equation tells you that the

y-coordinate of every point on

the line is –4. Draw the

horizontal line through (0, –4).

(0, –4)



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.





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

Example 3A: Classifying Pairs of Lines

y = 3x + 7, y = –3x – 4

The lines have different slopes, so they intersect.




Example 3B: Classifying Pairs of Lines

Solve the second equation for y to 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.




Example 3C: Classifying Pairs of Lines

2y – 4x = 16, y – 10 = 2(x - 1)

Solve both equations for y to 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 - 2

y = 2x + 8



Check It Out! Example 3

Determine whether the lines 3x + 5y = 2 and 3x + 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 to find the slope-intercept form.

3x + 5y = 2

5y = –3x + 2

3x + 6 = –5y



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



1 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.



2 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.



Solve 3

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.



The lines cross at (100, 135).

Both plans cost $135 for 100 miles.

Solve Continued 3



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 Back 4



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.



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

2 5

y + 1 = (x – 5)

2. the line through (5, –1)

with slope in point-slope

form.



Lesson Quiz: Part II


3. y – 3 = – x,

intersect

1 2

y – 5 = 2(x + 3)

4. 2y = 4x + 12, 4x – 2y = 8

parallel

Lines in the Coordinate PlaneLines in the Coordinate Plane

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