This document provides a lesson on lines in the coordinate plane. It includes examples of writing equations of lines in slope-intercept and point-slope form, graphing lines, and classifying pairs of lines as parallel, intersecting, or coinciding. It also contains a problem-solving application example involving writing equations to represent two car rental plans and finding where the lines intersect to determine the mileage where the plans' costs are equal.

1. 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

2. Holt Geometry
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

3. Holt Geometry
3-6 Lines in the Coordinate Plane
Graph lines and write their equations in
slope-intercept and point-slope form.
Classify lines as parallel, intersecting, or
coinciding.
Objectives

4. Holt Geometry
3-6 Lines in the Coordinate Plane
point-slope form
slope-intercept form
Vocabulary

5. Holt Geometry
3-6 Lines in the Coordinate Plane
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.

6. Holt Geometry
3-6 Lines in the Coordinate Plane

7. Holt Geometry
3-6 Lines in the Coordinate Plane
A line with y-intercept b contains the point (0, b).
A line with x-intercept a contains the point (a, 0).
Remember!

8. Holt Geometry
3-6 Lines in the Coordinate Plane
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.

9. Holt Geometry
3-6 Lines in the Coordinate Plane
Example 1B: Writing Equations In Lines
Write the equation of each line in the given form.
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.

10. Holt Geometry
3-6 Lines in the Coordinate Plane
Example 1C: Writing Equations In Lines
Write the equation of each line in the given form.
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

11. Holt Geometry
3-6 Lines in the Coordinate Plane
Check It Out! Example 1a
Write the equation of each line in the given form.
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.

12. Holt Geometry
3-6 Lines in the Coordinate Plane
Check It Out! Example 1b
Write the equation of each line in the given form.
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)

13. Holt Geometry
3-6 Lines in the Coordinate Plane
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

14. Holt Geometry
3-6 Lines in the Coordinate Plane
Graph each line.
Example 2B: Graphing Lines
y – 3 = –2(x + 4)
The equation is given in the
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

15. Holt Geometry
3-6 Lines in the Coordinate Plane
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)

16. Holt Geometry
3-6 Lines in the Coordinate Plane
Check It Out! Example 2a
Graph each line.
y = 2x – 3
The equation is given in the
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

17. Holt Geometry
3-6 Lines in the Coordinate Plane
Check It Out! Example 2b
Graph each line.
The equation is given in the
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
another point. Draw the line
containing the points.
(–2, 1)
run 3
rise –2

18. Holt Geometry
3-6 Lines in the Coordinate Plane
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)

19. Holt Geometry
3-6 Lines in the Coordinate Plane
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.

20. Holt Geometry
3-6 Lines in the Coordinate Plane

21. Holt Geometry
3-6 Lines in the Coordinate Plane
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.

22. Holt Geometry
3-6 Lines in the Coordinate Plane
Determine whether the lines are parallel,
intersect, or coincide.
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.

23. Holt Geometry
3-6 Lines in the Coordinate Plane
Determine whether the lines are parallel,
intersect, or coincide.
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

24. Holt Geometry
3-6 Lines in the Coordinate Plane
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

25. Holt Geometry
3-6 Lines in the Coordinate Plane
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

26. Holt Geometry
3-6 Lines in the Coordinate Plane
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.

27. Holt Geometry
3-6 Lines in the Coordinate Plane
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.

28. Holt Geometry
3-6 Lines in the Coordinate Plane
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.

29. Holt Geometry
3-6 Lines in the Coordinate Plane
The lines cross at
(100, 135).
Both plans cost $135
for 100 miles.
Solve Continued33

30. Holt Geometry
3-6 Lines in the Coordinate Plane
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

31. Holt Geometry
3-6 Lines in the Coordinate Plane
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.

32. Holt Geometry
3-6 Lines in the Coordinate Plane
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.

33. Holt Geometry
3-6 Lines in the Coordinate Plane
Lesson Quiz: Part II
Determine whether the lines are parallel,
intersect, or coincide.
3. y – 3 = – x,
intersect
1
2
y – 5 = 2(x + 3)
4. 2y = 4x + 12, 4x – 2y = 8
parallel