SlopeIntercept Form of a Line

3.4. SLOPE-INTERCEPT FORM OF A LINE
157
y
5
P (−1, 0)
−5
5
x
Δy = −3
Q(3, −3)
Δx = 4
−5
3.4
Slope-Intercept Form of a Line
1. First, compare y = 95 x − 6 with y = mx + b and note that m = 9/5 and
b = −6. Therefore, the slope of the line is 9/5 and the y-intercept is (0, −6).
Next, set up a coordinate system on a sheet of graph paper. Label
and scale each axis, then plot the y-intercept P (0, −6). Because the slope is
Δy/Δx = 9/5, start at the y-intercept P (0, −6), then move 9 units upward
and 5 units to the right, reaching the point Q(5, 3). Draw the line through the
points P and Q and label it with its equation y = 95 x − 6.
y
y = 95 x − 6
10
Δx = 5
Q(5, 3)
−10
10
Δy = 9
x
P (0, −6)
−10
3. First, compare y = − 11
4 x+ 4 with y = mx+ b and note that m = −11/4 and
b = 4. Therefore, the slope of the line is −11/4 and the y-intercept is (0, 4).
Next, set up a coordinate system on a sheet of graph paper. Label
and scale each axis, then plot the y-intercept P (0, 4). Because the slope is
Second Edition: 2012-2013
CHAPTER 3. INTRODUCTION TO GRAPHING
158
Δy/Δx = −11/4, start at the y-intercept P (0, 4), then move 11 units downward
and 4 units to the right, reaching the point Q(4, −7). Draw the line through
the points P and Q and label it with its equation y = − 11
4 x + 4.
y
10
P (0, 4)
−10
10
Δy = −11
x
Q(4, −7)
Δx = 4
x+4
y = − 11
4
−10
5. First, compare y = − 11
7 x+ 4 with y = mx+ b and note that m = −11/7 and
b = 4. Therefore, the slope of the line is −11/7 and the y-intercept is (0, 4).
Next, set up a coordinate system on a sheet of graph paper. Label
and scale each axis, then plot the y-intercept P (0, 4). Because the slope is
Δy/Δx = −11/7, start at the y-intercept P (0, 4), then move 11 units downward
and 7 units to the right, reaching the point Q(7, −7). Draw the line through
the points P and Q and label it with its equation y = − 11
7 x + 4.
y
10
P (0, 4)
−10
10
Δy = −11
x
Q(7, −7)
Δx = 7
−10
Second Edition: 2012-2013
x+4
y = − 11
7
3.4. SLOPE-INTERCEPT FORM OF A LINE
159
7. The y-intercept is (0, −7), so b = −7. Further, the slope is 9/5, so m = 9/5.
Substitute these numbers into the slope-intercept form of the line.
y = mx + b
9
y = x−7
5
Slope-intercept form.
Substitute: 9/5 for m, −7 for b.
Therefore, the slope-intercept form of the line is y = 95 x − 7.
Next, set up a coordinate system on a sheet of graph paper. Label
and scale each axis, then plot the y-intercept P (0, −7). Because the slope is
Δy/Δx = 9/5, start at the y-intercept P (0, −7), then move 9 units upward
and 5 units to the right, reaching the point Q(5, 2). Draw the line through the
points P and Q and label it with its equation y = 95 x − 7.
y
y = 95 x − 7
10
Δx = 5
Q(5, 2)
−10
10
x
Δy = 9
P (0, −7)
−10
9. The y-intercept is (0, −1), so b = −1. Further, the slope is 6/7, so m = 6/7.
Substitute these numbers into the slope-intercept form of the line.
y = mx + b
6
y = x−1
7
Slope-intercept form.
Substitute: 6/7 for m, −1 for b.
Therefore, the slope-intercept form of the line is y = 67 x − 1.
Next, set up a coordinate system on a sheet of graph paper. Label
and scale each axis, then plot the y-intercept P (0, −1). Because the slope is
Δy/Δx = 6/7, start at the y-intercept P (0, −1), then move 6 units upward
and 7 units to the right, reaching the point Q(7, 5). Draw the line through the
points P and Q and label it with its equation y = 67 x − 1.
Second Edition: 2012-2013
CHAPTER 3. INTRODUCTION TO GRAPHING
160
y
10
y = 67 x − 1
Δx = 7
Q(7, 5)
Δy = 6
−10
P (0, −1)
10
x
−10
11. The y-intercept is (0, −6), so b = −6. Further, the slope is 9/7, so m = 9/7.
Substitute these numbers into the slope-intercept form of the line.
y = mx + b
9
y = x−6
7
Slope-intercept form.
Substitute: 9/7 for m, −6 for b.
Therefore, the slope-intercept form of the line is y = 97 x − 6.
Next, set up a coordinate system on a sheet of graph paper. Label
and scale each axis, then plot the y-intercept P (0, −6). Because the slope is
Δy/Δx = 9/7, start at the y-intercept P (0, −6), then move 9 units upward
and 7 units to the right, reaching the point Q(7, 3). Draw the line through the
points P and Q and label it with its equation y = 97 x − 6.
y
10
y = 97 x − 6
Δx = 7
−10
Q(7, 3)
x
10
Δy = 9
P (0, −6)
−10
Second Edition: 2012-2013
3.4. SLOPE-INTERCEPT FORM OF A LINE
161
13. First, note that the y-intercept of the line (where it crosses the y-axis)
is the point P (0, 0). This means that b = 0 in the slope-intercept formula
y = mx + b.
Next, we need to determine the slope of the line. Try to locate a second
point on the line that passes directly through a lattice point, a point where a
horizontal and vertical gridline intersect. It appears that the point Q(3, −4)
qualifies.
y
6
Subtract the coordinates of P (0, 0)
from the coordinates of Q(3, −4) to
determine the slope:
P (0, 0)
−6
Δy
Δx
−4 − (0)
=
3−0
4
=−
3
m=
x
6
Q(3, −4)
−6
Alternate technique for finding the slope: Start at the y-intercept P (0, 0),
then move 4 units downward and 3 units to the right, reaching the point
Q(3, −4).
y
6
P (0, 0)
−6
6
x
Δy = −4
Q(3, −4)
Δx = 3
−6
This also indicates that the slope of the line is:
Δy
Δx
−4
=
3
m=
Second Edition: 2012-2013
CHAPTER 3. INTRODUCTION TO GRAPHING
162
Finally, substitute m = −4/3 and b = 0 in the slope-intercept form of
the line:
y = mx + b
4
y = − x + (0)
3
Hence, the equation of the line is y = − 34 x.
15. First, note that the y-intercept of the line (where it crosses the y-axis)
is the point P (0, 1). This means that b = 1 in the slope-intercept formula
y = mx + b.
Next, we need to determine the slope of the line. Try to locate a second
point on the line that passes directly through a lattice point, a point where a
horizontal and vertical gridline intersect. It appears that the point Q(4, −4)
qualifies.
y
6
Subtract the coordinates of P (0, 1)
from the coordinates of Q(4, −4) to
determine the slope:
P (0, 1)
−6
6
x
Q(4, −4)
Δy
Δx
−4 − 1
=
4−0
5
=−
4
m=
−6
Alternate technique for finding the slope: Start at the y-intercept P (0, 1),
then move 5 units downward and 4 units to the right, reaching the point
Q(4, −4).
Second Edition: 2012-2013
3.4. SLOPE-INTERCEPT FORM OF A LINE
163
y
6
P (0, 1)
−6
6
Δy = −5
x
Q(4, −4)
Δx = 4
−6
This also indicates that the slope of the line is:
Δy
Δx
−5
=
4
m=
Finally, substitute m = −5/4 and b = 1 in the slope-intercept form of
the line:
y = mx + b
5
y =− x+1
4
Hence, the equation of the line is y = − 54 x + 1.
17. First, note that the y-intercept of the line (where it crosses the y-axis)
is the point P (0, 0). This means that b = 0 in the slope-intercept formula
y = mx + b.
Next, we need to determine the slope of the line. Try to locate a second
point on the line that passes directly through a lattice point, a point where a
horizontal and vertical gridline intersect. It appears that the point Q(4, −3)
qualifies.
Second Edition: 2012-2013
CHAPTER 3. INTRODUCTION TO GRAPHING
164
y
6
Subtract the coordinates of P (0, 0)
from the coordinates of Q(4, −3) to
determine the slope:
P (0, 0)
−6
Δy
Δx
−3 − (0)
=
4−0
3
=−
4
m=
x
6
Q(4, −3)
−6
Alternate technique for finding the slope: Start at the y-intercept P (0, 0),
then move 3 units downward and 4 units to the right, reaching the point
Q(4, −3).
y
6
P (0, 0)
−6
6
Δy = −3
x
Q(4, −3)
Δx = 4
−6
This also indicates that the slope of the line is:
Δy
Δx
−3
=
4
m=
Finally, substitute m = −3/4 and b = 0 in the slope-intercept form of
the line:
y = mx + b
3
y = − x + (0)
4
Hence, the equation of the line is y = − 43 x.
Second Edition: 2012-2013
3.4. SLOPE-INTERCEPT FORM OF A LINE
165
19. First, note that the y-intercept of the line (where it crosses the y-axis)
is the point P (0, −4). This means that b = −4 in the slope-intercept formula
y = mx + b.
Next, we need to determine the slope of the line. Try to locate a second
point on the line that passes directly through a lattice point, a point where
a horizontal and vertical gridline intersect. It appears that the point Q(5, 0)
qualifies.
y
6
Subtract the coordinates of
P (0, −4) from the coordinates of
Q(5, 0) to determine the slope:
Δy
Δx
0 − (−4)
=
5−0
4
=
5
m=
x
6 0)
Q(5,
−6
P (0, −4)
−6
Alternate technique for finding the slope: Start at the y-intercept P (0, −4),
then move 4 units upward and 5 units to the right, reaching the point Q(5, 0).
y
6
Δx = 5
−6
x
6 0)
Q(5,
Δy = 4
P (0, −4)
−6
This also indicates that the slope of the line is:
Δy
Δx
4
=
5
m=
Second Edition: 2012-2013
CHAPTER 3. INTRODUCTION TO GRAPHING
166
Finally, substitute m = 4/5 and b = −4 in the slope-intercept form of
the line:
y = mx + b
4
y = x + (−4)
5
Hence, the equation of the line is y = 45 x − 4.
21. Set up a coordinate system. Label the horizontal axis with the time t (in
seconds) and the vertical axis with the velocity v (in meters per second). At
time t = 0, the initial velocity is 20 m/s. This gives us the point (0,20). The
velocity is increasing at a rate of 5 m/s each second. Let’s start at the point
(0,20), then move right 10 seconds and upward 50 m/s, arriving at the point
(10,70). Draw the line through these two points. Note that this makes the
slope Δv/Δt = (50 m/s)/(10 s), or 2 m/s per second.
v (m/s)
100
50
Δy = 50
(0, 20)
0
Δx = 10
0
10
20
t (s)
Because we know the intercept is (0,20) and the slope is 5, we use the
slope-intercept form to produce the equation of the line.
y = mx + b
Slope-intercept form.
y = 5x + 20
Substitute: 5 for m, 20 for b.
Replace y with v and x with t to produce the following result:
v = 5t + 20
To find the velocity at time t = 14 seconds, substitute 14 for t and simplify.
v = 5(14) + 20
v = 90
Thus, at t = 14 seconds, the velocity is v = 90 m/s.
Second Edition: 2012-2013
3.4. SLOPE-INTERCEPT FORM OF A LINE
167
23. Set up a coordinate system. Label the horizontal axis with the time t (in
minutes) and the vertical axis with the volume V (in gallons). At time t = 0,
the initial volume of water is 100 gallons. This gives us the point (0,100).
The volume of water is increasing at a rate of 25 gal/min. Start at the point
(0,100), then move right 10 minutes and upward 250 gallons, arriving at the
point (10,350). Draw the line through these two points. Note that this makes
the slope ΔV /Δt = (250 gal)/(10 min), or 25 gal/min, as required.
v (gal)
1,000
500
Δy = 250
(0, 100)
0
Δx = 10
0
10
20
t (min)
Because we know the intercept is (0,100) and the slope is 25, we use the
slope-intercept form to produce the equation of the line.
y = mx + b
Slope-intercept form.
y = 25x + 100
Substitute: 25 for m, 100 for b.
Replace y with V and x with t to produce the following result:
V = 25t + 100
To find the time it takes the volume of water to reach 400 gallons, substitute
400 for V and solve for t.
400 = 25t + 100
400 − 100 = 25t + 100 − 100
300 = 25t
300
25t
=
25
25
12 = t
Substitute 400 for V .
Subtract 100 from both sides.
Simplify.
Divide both sides by 25.
Simplify.
Thus, it takes 12 minutes for the volume of water to reach 400 gallons.
Second Edition: 2012-2013