4-7
Point-Slope Form
Connection: Relating Slope-Intercept,
Point-Slope, and Standard Forms
Essential question: What properties of linear functions does each linear function
form illustrate?
TEACH
Standards for
Mathematical Content
1
F-IF.3.8 Write a function defined by an expression
in different but equivalent forms to reveal and
explain different properties of the function.
F-LE.1.2 Construct linear … functions … given …
two input-output pairs (include reading these from
a table).*
EXAMPLE
Questioning Strategies
• Would you use standard form or the slopeintercept form to find the x-intercept? Explain.
Standard form; to find the x-intercept, substitute
0 for y, and solve the equation for x.
• Could you use standard form to graph the
equation in part A? Explain. Yes; you can use
Prerequisites
Identifying Linear Functions
Using Intercepts
The Slope Formula
Slope-Intercept Form
standard form to find the x- and y-intercepts and
use those to graph the equation.
EXTRA EXAMPLE
Write an equation for 9x - 6y = 18 in slopeintercept form. Then use that form to identify the slope and y-intercept. y = _ 32   x - 3; slope:  _32 , 
Math Background
Students have learned three different forms for
writing linear equations. In standard form, Ax + By = C where A, B, and C are real numbers
and A and B are not both 0. In slope-intercept form,
y = mx + b where m is the slope of the line and b is
the y-intercept. In point-slope form,
y - y1 = m(x - x1 ) where m is the slope of the line
and (x1 , y1 ) is a point on the line.
y-intercept: -3
2
EXAMPLE
No; however, using standard form will take fewer
steps than using point-slope form directly to find
the x- and y-intercepts.
INTR O D U C E
• Would you use the standard form or the pointslope form to graph the equation? Explain. Point-
Tell students they will be given a linear equation
to graph. Ask them what information they need to
know about the line in order to graph it. Make a list
of the students’ responses on the board. (The list
should include slope, y-intercept, x-intercept, and
points on the line.) Then tell students they can pick
the form of the equation (standard, slope-intercept,
or point-slope). Ask students which form they
would pick and why.
slope form; the slope and a point on the line are
easily seen in the point-slope form, and the slope
and a point on the line are used to graph the line.
EXTRA EXAMPLE
Write an equation for y - 6 = -2(x - 1) in standard
form. Then use that form to identify the x- and
y-intercepts. 2x + y = 8; x-intercept: 4, y-intercept: 8
Avoid Common Errors
When multiplying to eliminate a fraction in an
equation, remind students to multiply both sides of
the equation by the number, not just the side of the
equation with the fraction. Similarly, when dividing
a variable term by a number to isolate a variable,
remind students to divide every term on both sides
of the equation by that number.
Chapter 4 225
Lesson 7
© Houghton Mifflin Harcourt Publishing Company
Questioning Strategies
• Do you have to use the standard form of the
equation to find the x- and y-intercepts? Explain.
Name
Class
Notes
4-7
Date
Point-Slope Form
Connection: Relating Slope-Intercept, Point-Slope,
and Standard Forms
Essential question: What properties of linear functions does each linear function
form illustrate?
F-IF.3.8
1
EXAMPLE
Writing Equations in Slope-Intercept Form
Write an equation for -4x + 5y = 10 in slope-intercept form. Then use that form
to identify the slope and y-intercept.
Rewrite the equation in slope-intercept form by solving for y.
A
-4x + 5y = 10
Standard form of the equation
5y = 4x + 10
y=
__4 x
5
10
+ __
5
4x + 2
y = __
5
B
Add 4x to each side.
Divide each side by 5.
Simplify.
Identify the slope and y-intercept of the line.
The equation is in the form y = mx + b, where m is the slope and b is the
y-intercept. So, the slope of the line is
__4
5
and the y-intercept is
2 .
© Houghton Mifflin Harcourt Publishing Company
REFLECT
1a. In the Example, which equation form would you use to graph the equation?
Explain.
Slope-intercept form; the slope and y-intercept are most helpful when graphing a
line, and the slope and y-intercept are easily determined when the equation is in
slope-intercept form.
1b. Rewrite Ax + By = C in slope-intercept form. Explain how you can use this form to
identify the slope and y-intercept of a line when its equation is given in standard
form.
A
A
y = - __
x + _C ; in the rewritten equation, - __
represents the slope of the line and _C
B
B
B
B
represents the y-intercept of the line. For a linear equation given in standard form,
C
A
calculate - __
for the slope and _
for the y-intercept.
B
B
Chapter 4
225
Lesson 7
F-IF.3.8
2
EXAMPLE
Writing Equations in Standard Form
2 (x - 9) in standard form. Then use that form to
Write an equation for y - 4 = __
3
identify the x- and y-intercepts.
Rewrite the equation in standard form by collecting the x- and y-terms on one side of
the equation.
y - 4 = __23 (x - 9)
3(y - 4) =
3
r__23 (x - 9)
3y - 12 = 2x - 18
3y = 2x - 6
-2x + 3y = -6
B
Multiply each side by 3.
Simplify.
Add 12 to each side.
Subtract 2x from each side.
Find the x-intercept by substituting 0 for y and solving for x.
(
-2x + 3
0
) = -6
-2x = -6
x=
The x-intercept is
C
Point-slope form of the equation
3
Substitute 0 for y.
Simplify.
Solve for x.
3 .
Find the y-intercept by substituting 0 for x and solving for y.
(
-2
0
) + 3y = -6
3y = -6
y = -2
Substitute 0 for x.
Simplify.
Solve for y.
The y-intercept is -2 .
REFLECT
2a. Find the x-intercept of the line in the Example by substituting 0 for y and solving for
x in the point-slope form of the equation. Is it easier to find the x-intercept using
the standard form or the point-slope form? Explain your answer.
Sample answer: The x-intercept is 3; standard form; there are fewer mathematical
calculations needed to solve for x when using standard form.
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
A
2b. Explain why you substitute 0 for x in the equation to find the y-intercept.
The y-intercept of the graph of a line is where the graph crosses the y-axis. The
point that corresponds to the y-intercept b is (0, b). To find the value of b, you
have to substitute 0 for x in the equation and solve for y.
Chapter 4
Chapter 4
226
Lesson 7
226
Lesson 7
3
Summarize
Have students complete the table by stating which
form of a linear equation they would use to write an
equation for the line with the given information.
EXAMPLE
Questioning Strategies
• In Part C, what is the point-slope form if you
substitute (-4, 9)? How can you show that this
equation is equivalent to the point-slope equation
using (8, 6)? y - 9 = -__14(x + 4); Change both
point-slope equations to either standard form or
slope-intercept form.
• Suppose the slope and a point on the line
were given. Which equation form would you
use to write an equation for the line? Explain.
If the point were the y-intercept, I would use
slope-intercept form because the slope and the
y-intercept can be substituted directly into this
form. If the point were not the y-intercept,
I would use point-slope form because the slope
and the point can be substituted directly into this
form.
Given Information
Equation Form
slope and y-intercept
slope-inter.
two points on the line
point-slope
a horizontal line and one
point on the line
standard
slope and point (x, y) on
the line where x ≠ 0
point-slope
a vertical line and one
point on the line
standard
slope and point (x, y) on
the line where x = 0
slope-inter.
Highlighting
the Standards
EXTRA EXAMPLE
Determine which form of a linear equation to use to
write an equation for the line with a slope of __45 that
passes through (0, -5). Then write an equation for
the line. slope-intercept form; y = _45 x - 5
Teaching Strategy
Before beginning 3 EXAMPLE , have students
read the summary table at the bottom of the page.
Discuss how this information is helpful in choosing
an appropriate equation form when writing a linear
equation. To reinforce students’ understanding, ask
general questions such as “Which form will you use
to write a linear equation given the slope and the
y-intercept of the line?”
CLOSE
Essential Question
What properties of linear functions does each linear
function form illustrate?
PR ACTICE
Slope-intercept form shows the slope and the
y-intercept of the line. Point-slope form shows the
slope and a point that lies on the line. Standard
form shows whether a line is vertical or horizontal
and can be used to find the x- and y-intercepts of
lines that are not horizontal or vertical.
Where skills are
taught
Where skills are
practiced
1 EXAMPLE
EXS. 2, 3
2 EXAMPLE
EXS. 1, 4
3 EXAMPLE
EXS. 5-8
Exercise 9: Students determine whether it is easier
to transform an equation given in standard form to
slope-intercept form or to point-slope form.
Exercise 10: Students write formulas for finding the
x- and y-intercepts of any line given in standard form.
Chapter 4
227
Lesson 7
© Houghton Mifflin Harcourt Publishing Company
Exercises 1-4 provide opportunities to
address Mathematical Practices Standard
2 (Reason abstractly and quantitatively)
and Standard 7 (Look for and make use
of structure). Using properties of equality,
students manipulate a linear equation so that
it is written in a specific form. Then students
analyze the terms in the rewritten equation.
These terms illustrate characteristics of the
line, such as its slope or y-intercept. Students
should understand that transforming a linear
equation does not change the characteristics
of the line—it changes the form of the
equation so that the characteristics of the line
are easily determined.
Notes
F-LE.1.2
3
EXAMPLE
Choosing an Appropriate Form of a Linear Equation
Determine which form of a linear equation to use to write an equation for the
line that passes through (-4, 9) and (8, 6). Then write an equation for the line.
A
To determine which form to use, identify the given information.
Two points on the line are given, and neither of them is the y-intercept. So, write the
point-slope
equation using
B
9
6-
y -y
2
1
____________ =
m = ______
x -x =
2
C
form.
Find the slope of the line using the two points.
8
1
- (-4)
-3
___
12
y - y1 = m(x - x1)
6
__1
=4
4
point-slope
Write the equation in
y-
1
= - __
form using the point (8, 6).
General form of the equation
(x -
8
)
Substitute values for x1, y1, and m.
REFLECT
3a. Could (-4, 9) have been used to write an equation of the line in Part C? Explain.
Yes; both given points are on the line so either point can be used to write an
equation for the line.
3b. Suppose you are given the y-intercept and the slope of a line. Which linear
equation form would you use to write an equation for the line? Explain your
reasoning.
© Houghton Mifflin Harcourt Publishing Company
Slope-intercept form; the given information can be substituted in the general
slope-intercept form and no calculations need to be made.
The following table provides a summary of the information presented in the Examples
above.
Form
Equation
Information
When to Use
slope-intercept
y = mx + b
· m is the slope.
· b is the y-intercept.
· When given the slope and
the y-intercept
point-slope
y - y1 = m(x - x1)
· m is the slope.
· (x1, y1) lies on the line.
· When given the slope and
one point on the line
· When given two points on
the line
standard
Ax + By = C
· A, B, and C are real numbers.
· A and B are not both 0.
· When given a horizontal or
vertical line and one point
on the line
Chapter 4
227
Lesson 7
PRACTICE
Rewrite the equation to find the characteristics of the line.
1. Rewrite y = -__32 x + 6 in standard form. Identify the x-intercept of the line.
7
7
x + 2; the slope is _
. The y-intercept is 2.
y=_
9
9
3. Rewrite y + 1 = 4(x + 3) in slope-intercept form. Identify the y-intercept of the
line.
y = 4x + 11; the y-intercept is 11.
4. Rewrite y + 25 = -__53(x - 12) in standard form. Identify the x- and y-intercepts of
the line.
5x + 3y = -15; the x-intercept is -3. The y-intercept is -5.
Determine which form of a linear equation to use to write an equation for the
line with the given characteristics. Then write an equation for the line in that
form.
5. passes through (-5, -4), (7, 5)
3
(x + 5)
point-slope form; y + 4 = _
6. slope = -__27 ; y-intercept = 9
2
slope-intercept form; y = -_
x+9
7
4
11 ; passes through (-3, 6)
8. slope = __
13
7. vertical line through (8, 1)
11
(x + 3)
point-slope form; y - 6 = __
standard form; x = 8
13
9. You are asked to rewrite -9x + 4y = 14 to find the slope of the line. Would you
rewrite the equation in slope-intercept form or point-slope form? Explain.
Possible answer: Slope-intercept form; writing the equation in slope-intercept form
will take fewer steps than writing the equation in point-slope form.
10. Find the x- and y-intercepts of the graph of Ax + By = C. Explain how you can use
these intercepts to find the x- and y-intercepts of any line in standard form.
C
C
The x-intercept is __
, and the y-intercept is _
; for an equation written in standard
A
B
C
C
form, substitute values in __
to find the x-intercept of the line and in _
to find the
A
B
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
3x + 2y = 12; the x-intercept is 4.
2. Rewrite -7x + 9y = 18 in slope-intercept form. Identify the slope and y-intercept
of the line.
y-intercept of the line.
Chapter 4
Chapter 4
228
Lesson 7
228
Lesson 7
Problem Solving
ADD I T I O NA L P R AC TI C E
AND PRO BL E M S O LV I N G
1. Possible answer:
y - 130 = 1.2(x - 10);
y = 1.2x + 118;
136
Assign these pages to help your students practice
and apply important lesson concepts. For
additional exercises, see the Student Edition.
2. y = 3x + 32
Answers
1 x + 5; $7.50
3. y = _
10
Additional Practice
4. B
1. y - 2 = 3(x + 4)
2. y + 1 = -(x - 6)
5. F
6. D
3.
4.
7. y = 4x - 7
9. x-int: 1, y-int: -2
© Houghton Mifflin Harcourt Publishing Company
5. y = -4x + 1
1 x-1
6. y = _
2
1 x-3
8. y = _
2
10. x-int:6, y-int: 8
11. y = 0.17x + 3; $13.20
Chapter 4
229
Lesson 7
Name
Class
Notes
4-7
Date
Additional Practice
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Problem Solving
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Chapter 4
Chapter 4
230
Lesson 7
230
Lesson 7