Writing Equations in Point

4.7
Writing Equations in
Point-Slope Form
How can you write an equation of a line when
you are given the slope and a point on the line?
1
ACTIVITY: Writing Equations of Lines
Work with a partner.
●
Sketch the line that has the given slope and passes through
the given point.
●
Find the y-intercept of the line.
●
Write an equation of the line.
1
3
a. m = −2
b. m = —
y
y
8
6
7
5
6
4
5
3
4
2
3
1
2
Ź5 Ź4 Ź3 Ź2 Ź1
Ź1
1
Ź6 Ź5 Ź4 Ź3 Ź2 Ź1
Ź1
1
2
3
4
5
Writing Equations
In this lesson, you will
● write equations of lines
using a slope and a point.
● write equations of lines
using two points.
Preparing for Standard
8.F.4
184
Chapter 4
3
4
5
6
7 x
1
2
3
4
5
6 x
Ź3
Ź2
Ź4
Ź3
Ź5
Ź4
Ź6
5
2
c. m = −—
COMMON
CORE
2
Ź2
6 x
2
3
1
d. m = —
y
y
6
7
5
6
4
5
3
4
2
3
1
2
Ź5 Ź4 Ź3 Ź2 Ź1
Ź1
1
2
3
4
5
6
7 x
1
Ź2
Ź6 Ź5 Ź4 Ź3 Ź2 Ź1
Ź1
Ź3
Ź2
Ź4
Ź3
Ź5
Ź4
Ź6
Ź5
Graphing and Writing Linear Equations
2
ACTIVITY: Deriving an Equation
Work with a partner.
y
a. Draw a nonvertical line that passes through the
point (x1, y1).
(x1, y1)
b. Plot another point on your line. Label this point
as (x, y). This point represents any other point on
the line.
Math
Practice
Construct
Arguments
O
x
c. Label the rise and the run of the line through
the points (x1, y1) and (x, y).
How does a graph
help you derive
an equation?
d. The rise can be written as y − y1. The run can be written as x − x1.
Explain why this is true.
e. Write an equation for the slope m of the line using the expressions
from part (d).
f.
3
Multiply each side of the equation by the expression in the
denominator. Write your result. What does this result represent?
ACTIVITY: Writing an Equation
Work with a partner.
Savings Account
For 4 months, you saved $25 a month. You
now have $175 in your savings account.
A
250
●
Draw a graph that shows the balance
in your account after t months.
Use your result from Activity 2 to
write an equation that represents
the balance A after t months.
Balance (dollars)
225
●
200
175
150
125
100
75
50
25
0
0
1
2
3
4
5
6
7
8
9 t
Time (months)
4. Redo Activity 1 using the equation you found in Activity 2. Compare the
results. What do you notice?
5. Why do you think y − y1 = m(x − x1) is called the point-slope form of
the equation of a line? Why do you think it is important?
6. IN YOUR OWN WORDS How can you write an equation of a line when
you are given the slope and a point on the line? Give an example that is
different from those in Activity 1.
Use what you learned about writing equations using a slope
and a point to complete Exercises 3 – 5 on page 188.
Section 4.7
Writing Equations in Point-Slope Form
185
4.7
Lesson
Lesson Tutorials
Key Vocabulary
point-slope form,
p. 186
Point-Slope Form
A linear equation written in the form y − y1 = m(x − x1)
is in point-slope form. The line passes through the point
(x1, y1), and the slope of the line is m.
Words
y
slope
(x, y)
y Ź y1
y − y1 = m(x − x1)
Algebra
(x1, y1)
x Ź x1
passes through (x1, y1)
EXAMPLE
1
O
x
Writing an Equation Using a Slope and a Point
Write in point-slope form an equation of the line that passes through
2
3
the point (−6, 1) with slope —.
y − y1 = m(x − x1)
Write the point-slope form.
2
3
Substitute — for m, −6 for x1, and 1 for y1.
2
3
Simplify.
2
3
y − 1 = —[x − (−6)]
y − 1 = —(x + 6)
2
3
So, the equation is y − 1 = — (x + 6).
Check Check that (−6, 1) is a solution of the equation.
2
3
y − 1 = —(x + 6)
Write the equation.
? 2
1 − 1 = —(−6 + 6)
Substitute.
3
0=0
Exercises 6 – 11
Chapter 4
Simplify.
Write in point-slope form an equation of the line that passes through
the given point and has the given slope.
1. (1, 2); m = −4
186
✓
2.
(7, 0); m = 1
Graphing and Writing Linear Equations
3.
3
4
(−8, −5); m = −—
EXAMPLE
2
Writing an Equation Using Two Points
Write in slope-intercept form an equation of the line that passes
through the points (2, 4) and (5, −2).
Study Tip
y −y
x2 − x1
You can use either of
the given points to
write the equation of
the line.
Use m = −2 and
(5, −2).
y − (−2) = −2(x − 5)
y + 2 = −2x + 10
y = −2x + 8
−6
3
Then use the slope m = −2 and the point (2, 4) to write an
equation of the line.
✓
y − y1 = m(x − x1)
Write the point-slope form.
y − 4 = −2(x − 2)
Substitute −2 for m, 2 for x1, and 4 for y1.
y − 4 = −2x + 4
Distributive Property
y = −2x + 8
EXAMPLE
−2 − 4
5−2
2
1
Find the slope: m = —
= — = — = −2
3
Write in slope-intercept form.
Real-Life Application
You finish parasailing and are being pulled back to the boat. After
2 seconds, you are 25 feet above the boat. (a) Write and graph an
equation that represents your height y (in feet) above the boat after
x seconds. (b) At what height were you parasailing?
a. You are being pulled down at the rate of 10 feet per second.
So, the slope is −10. You are 25 feet above the boat after 2 seconds.
So, the line passes through (2, 25). Use the point-slope form.
10 feet
per
second
y − 25 = −10(x − 2)
Substitute for m, x1, and y1.
y − 25 = −10x + 20
Distributive Property
y = −10x + 45
Write in slope-intercept form.
So, the equation is y = −10x + 45.
b. You start descending when x = 0. The
y-intercept is 45. So, you were parasailing
at a height of 45 feet.
y
45
y â Ź10x à 45
40
35
30
(2, 25)
25
20
15
10
5
0
Exercises 12 – 17
0
1
2
3
4
5
6
7 x
Write in slope-intercept form an equation of the line that passes
through the given points.
4. (−2, 1), (3, −4)
5.
(−5, −5), (−3, 3)
6.
(−8, 6), (−2, 9)
7. WHAT IF? In Example 3, you are 35 feet above the boat after
2 seconds. Write and graph an equation that represents your
height y (in feet) above the boat after x seconds.
Section 4.7
Writing Equations in Point-Slope Form
187
Exercises
4.7
Help with Homework
1. VOCABULARY From the equation y − 3 = −2(x + 1), identify the slope and a
point on the line.
2. WRITING Describe how to write an equation of a line using (a) its slope and a
point on the line and (b) two points on the line.
6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-
Use the point-slope form to write an equation of the line with the given slope
that passes through the given point.
1
2
3
4
3. m = —
4. m = −—
5. m = −3
y
y
4
y
4
3
3
3
2
2
2
1
1
1
Ź4 Ź3 Ź2 Ź1
Ź1
1
2
Ź5 Ź4 Ź3 Ź2 Ź1
Ź1
3 x
1
2 x
Ź2 Ź1
Ź1
1
2
3
4
5 x
Ź2
Ź2
Ź2
Ź3
Ź3
Ź3
Ź4
Write in point-slope form an equation of the line that passes through the given
point and has the given slope.
1
2
3
3
4
6. (3, 0); m = −—
7. (4, 8); m = —
1
7
5
3
9. (7, −5); m = −—
10. (3, 3); m = —
8. (1, −3); m = 4
11. (−1, −4); m = −2
Write in slope-intercept form an equation of the line that passes through the given points.
2 12. (−1, −1), (1, 5)
15. (4, 1), (8, 2)
13. (2, 4), (3, 6)
14. (−2, 3), (2, 7)
16. (−9, 5), (−3, 3)
17. (1, 2), (−2, −1)
18. CHEMISTRY At 0 °C, the volume of a gas is 22 liters. For each degree the
temperature T (in degrees Celsius) increases, the volume V (in liters) of the
2
25
gas increases by —. Write an
equation that represents the
volume of the gas in terms of
the temperature.
188
Chapter 4
Graphing and Writing Linear Equations
19. CARS After it is purchased, the value of a new car decreases $4000 each year.
After 3 years, the car is worth $18,000.
a. Write an equation that represents the value V (in dollars) of the
car x years after it is purchased.
b. What was the original value of the car?
20. REASONING Write an equation of a line that passes through the point (8, 2) that
is (a) parallel and (b) perpendicular to the graph of the equation y = 4x − 3.
21. CRICKETS According to Dolbear’s law, you can predict the
temperature T (in degrees Fahrenheit) by counting the
number x of chirps made by a snowy tree cricket in 1 minute.
For each rise in temperature of 0.25°F, the cricket makes an
additional chirp each minute.
a. A cricket chirps 40 times in 1 minute when the temperature
is 50°F. Write an equation that represents the temperature in terms of
the number of chirps in 1 minute.
b. You count 100 chirps in 1 minute. What is the temperature?
c. The temperature is 96 °F. How many chirps would you expect the
cricket to make?
Leaning Tower of Pisa
y
22. WATERING CAN You water the plants in your classroom at
a constant rate. After 5 seconds, your watering can contains
58 ounces of water. Fifteen seconds later, the can contains
28 ounces of water.
(10.75, 42)
a. Write an equation that represents the amount y (in ounces)
of water in the can after x seconds.
b. How much water was in the can when you started
watering the plants?
c. When is the watering can empty?
23.
Problem
The Leaning Tower of Pisa in Italy was built
Solving
between 1173 and 1350.
a. Write an equation for the yellow line.
x
7.75 m
b. The tower is 56 meters tall. How far off center is the
top of the tower?
Graph the linear equation. (Section 4.4)
24. y = 4x
25. y = −2x + 1
26. y = 3x − 5
27. MULTIPLE CHOICE What is the x-intercept of the equation 3x + 5y = 30?
(Section 4.5)
A −10
○
B −6
○
Section 4.7
C 6
○
D 10
○
Writing Equations in Point-Slope Form
189