Basics
(0, b) = y-intercept = slope-intercept = point where line crosses y-axis
A question involving slope-intercept will often require you to use two points to calculate the slope and y-intercept on the way to writing the equation of a line.
Although plotting points and a line on an x-y coordinate plane or grid can help visualize the process, a question involving slope-intercept can often be solved without graphing.
Formulas (Mathematics Formula Sheet)
slope-intercept form of the equation of a line
y = mx + b
slope of a line
m =
Formulas (NOT on Mathematics Formula Sheet)
slope-intercept b
b = y – mx
Question
Using the points (1, 4) and (3, 9) in the graph below, write the slope-intercept equation for Line L. The slope and y-intercept should be in decimal form.
Answer
y = 2.5x + 1.5
Answer Process
(x1, y1) = (1, 4)
(x2, y2) = (3, 9)
slope of a line
m =
m = =
= 2.5 after toggle
slope-intercept b
b = y – mx
(x1, y1) or (x2, y2) can be used to calculate b; in this case, (x1, y1) = (1, 4) will be used:
slope-intercept b
b = y – mx = 4 – 2.5 × 1 = 1.5
slope-intercept form of the equation of a line
y = mx + b
y = 2.5x + 1.5
Input | Display | Comment |
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blinker | clears screen |
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m = |
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2.5 | m (Toggle) |
4 – 2.5 × 1 | 4-2.5*1 | b = y – mx |
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1.5 | b |
Slope-Intercept
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Question
Using the points (-2, 3) and (-4, 7) in the graph below, write the slope-intercept equation for Line L. The slope and y-intercept should be in decimal form.
Answer
y = -2x – 1
Answer Process
(x1, y1) = (-2, 3)
(x2, y2) = (-4, 7)
slope of a line
m =
m = =
= -2
slope-intercept b
b = y – mx
(x1, y1) or (x2, y2) can be used to calculate b; in this case, (x1, y1) = (-2, 3) will be used:
b = y – mx = 3 – -2 × -2 = -1
slope-intercept form of the equation of a line
y = mx + b
y = -2x – 1
Input | Display | Comment |
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blinker | clears screen |
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m = |
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-2 | m |
3 – -2 × -2 | 3- -2* -2 | b = y – mx |
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-1 | b |
Slope-Intercept
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Question
Without a graph, write the slope-intercept equation for the line passing through points (3, 7) and (5, 9.5). The slope and y-intercept should be in decimal form.
Answer
y = 1.25x + 3.25
Answer Process
(x1, y1) = (3, 7)
(x2, y2) = (5, 9.5)
slope of a line
m =
m = = 1.25
slope-intercept b
b = y – mx
(x1, y1) or (x2, y2) can be used to calculate b; in this case, (x1, y1) = (3, 7) will be used:
b = y – mx = 7 – 1.25 × 3 = 3.25
slope-intercept form of the equation of a line
y = mx + b
y = 1.25x + 3.25
Input | Display | Comment |
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blinker | clears screen |
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m = |
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1.25 | m |
7 – 1.25 × 3 | 7-1.25*3 | b = y – mx |
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3.25 | b |
Slope-Intercept
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Question
Without a graph, write the slope-intercept equation for the line passing through points (-3, 6) and (-5, 8.5). The slope and y-intercept should be in decimal form.
Answer
y = -1.25x + 2.25
Answer Process
(x1, y1) = (-3, 6)
(x2, y2) = (-5, 8.5)
slope of a line
m =
m = = -1.25
slope-intercept b
b = y – mx
(x1, y1) or (x2, y2) can be used to calculate b; in this case, (x1, y1) = (-3, 6) will be used:
b = y – mx = 6 – 1.25 × -3 = 2.25
slope-intercept form of the equation of a line
y = mx + b
y = -1.25x + 2.25
Input | Display | Comment |
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blinker | clears screen |
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m = |
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-1.25 | m |
6 – -1.25 × -3 | 6- -1.25* -3 | b = y – mx |
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2.25 | b |
Slope-Intercept
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Question
The following table shows the age in years and height in inches of a boy and a girl.
Write the slope-intercept equation for the line passing through the boy’s growth points at (2, 34.00) and (4, 40.00).
Boy | Girl | |||
Age (yr) | Height (in) | Age (yr) | Height (in) | |
2 | 34.00 | 2 | 33.50 | |
3 | 37.50 | 3 | 37.00 | |
4 | 40.00 | 4 | 39.50 | |
5 | 43.00 | 5 | 42.25 | |
6 | 45.25 | 6 | 45.00 | |
7 | 48.00 | 7 | 48.00 | |
8 | 50.25 | 8 | 50.00 | |
9 | 52.50 | 9 | 52.50 | |
10 | 54.50 | 10 | 54.25 |
Answer
y = 3x + 28
Answer Process
(x1, y1) = (2, 34.00)
(x2, y2) = (4, 40.00)
slope of a line
m =
m = = 3
slope-intercept b
b = y – mx
(x1, y1) or (x2, y2) can be used to calculate b; in this case, (x1, y1) = (2, 34.00) will be used:
b = y – mx = 34.00 – 3 × 2 = 28
slope-intercept form of the equation of a line
y = mx + b
y = 3x + 28
Input | Display | Comment |
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blinker | clears screen |
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m = |
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3 | m |
34.00 – 3 × 2 | 34.00-3*2 | b = y – mx |
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28 | b |
Slope-Intercept
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Question
The following table shows the growth of an investment over time.
Time (years) | Value (dollars) |
2 | 100 |
4 | 120 |
6 | 140 |
8 | 160 |
10 | 180 |
Based on the data in the table, which of the following slope-intercept equations represents the growth of an investment over time, where x is the year and y is the dollar value?
A. y = -10x + 80
B. y = 10x – 80
C. y = -10x – 80
D. y = 10x + 80
Answer
D. y = 10x + 80
Answer Process
Any two points in the data table will do; in this case, the following points will be used:
(x1, y1) = Point (2, 100)
(x2, y2) = Point (4, 120)
slope of a line
m =
m = = 10
slope-intercept b
b = y – mx
(x1, y1) or (x2, y2) can be used to calculate b; in this case, (x1, y1) = (2, 100) will be used:
b = y – mx = 100 – 10 × 2 = 80
slope-intercept form of the equation of a line
y = mx + b
y = 10x + 80
Input | Display | Comment |
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blinker | clears screen |
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m = |
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10 | m |
100 – 10 × 2 | 100-10*2 | b = y – mx |
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80 | b |
Slope-Intercept
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Question
Bob planted some prairie grass to beautify a corner of his yard. After 3 weeks, the grass was 12 inches tall. After 7 weeks, the grass was 24 inches tall. If it grows at the same rate, how tall will the grass be at week 10?
Answer
33 inches
Answer Process
(x1, y1) = (3, 12)
(x2, y2) = (7, 24)
slope of a line
m =
m = = 3
slope-intercept b
b = y – mx
(x1, y1) or (x2, y2) can be used to calculate b; in this case, (x1, y1) = (3, 12) will be used:
b = y – mx = 12 – 3 × 3 = 3
slope-intercept form of the equation of a line
y = mx + b
y = 3x + 3
See Plug-In.
Plug in 10 for x.
y = 3 × 10 + 3 = 33
Input | Display | Comment |
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blinker | clears screen |
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m = |
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3 | m |
12 – 3 × 3 | 12-3*3 | b = y – mx |
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3 | b |
3 × 10 + 3 | 3*10+3 | y = mx + b Plug in 10 for x. |
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33 | Answer |
Slope-Intercept
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Practice – Questions
1. Using the points (1, 6.5) and (2, 10) in the graph below, write the slope-intercept equation for Line L. The slope and y-intercept should be in decimal form.
2. Without a graph, write the slope-intercept equation for the line passing through points (-4, 8) and (-8, 13). The slope and y-intercept should be in decimal form.
3. The following table shows the age in years and height in inches of a boy and a girl.
Write the slope-intercept equation for the line passing through the girl’s growth points at (3, 37.00) and (7, 48.00). The slope and y-intercept should be in decimal form.
Boy | Girl | |||
Age (yr) | Height (in) | Age (yr( | Height (in) | |
2 | 34.00 | 2 | 33.50 | |
3 | 37.50 | 3 | 37.00 | |
4 | 40.00 | 4 | 39.50 | |
5 | 43.00 | 5 | 42.25 | |
6 | 45.25 | 6 | 45.00 | |
7 | 48.00 | 7 | 48.00 | |
8 | 50.25 | 8 | 50.00 | |
9 | 52.50 | 9 | 52.50 | |
10 | 54.50 | 10 | 54.25 |
4. The following table shows the growth of an investment over time.
Time (years) | Value (dollars) |
3 | 250 |
5 | 500 |
7 | 750 |
9 | 1000 |
11 | 1250 |
Based on the data in the table, which of the following slope-intercept equations represents the growth of an investment over time, where x is the year and y is the dollar value?
A. y = -125x – 125
B. y = -125x + 125
C. y = 125x – 125
D. y = 125x + 125
5. The following table shows the depreciation (loss in value) of an automobile over time.
Age (years) | Value (dollars) |
0 | 25000 |
2 | 18000 |
4 | 11000 |
6 | 4000 |
Based on the data in the table, which of the following slope-intercept equations represents the depreciation (loss in value) of an automobile over time, where x is the age in years and y is the dollar value?
A. y = 3500x – 25000
B. y = -3500x + 25000
C. y = -3500x – 25000
D. y = 3500x + 25000
6. Bob planted some prairie grass to beautify a corner of his yard. After 4 weeks, the grass was 18 inches tall. After 8 weeks, the grass was 48 inches tall. If it grows at the same rate, how tall will the grass be at week 12?
7. Betty sent 2 text messages at 4:00 AM and 7 text messages at 6:00 AM. If Betty increases her text message output at the same rate, how many texts will she send at 10 AM?
8. Bob spent $3.00 on popcorn at 5:00 PM and $4.50 on popcorn at 7:00 PM. If Bob increases his popcorn expenditures at the same rate, how much will he spend on popcorn at 10:00 PM?
9. The following table shows the approximate weight in kilograms (kg) over the first ten weeks of a pig’s life.
Age (weeks) | Weight (kg) |
0 | 1 |
1 | 2 |
2 | 3 |
3 | 5 |
4 | 8 |
5 | 10 |
6 | 15 |
7 | 18 |
8 | 20 |
9 | 25 |
10 | 30 |
Using the table’s data points of (0, 1) and (2, 3), write a slope-intercept equation for the pig’s gain in weight, where x is the age in weeks and y is the weight in kg.
10. With reference to Question 9, use the table’s data points of (8, 20) and (10, 30) to write a slope-intercept equation for the pig’s gain in weight, where x is the age in weeks and y is the weight in kg.
Practice – Answers
1. y = 3.5x + 3
2. y = -1.25x + 3
3. y = 2.75x + 28.75
4. C. y = 125x – 125
5. B. y = -3500x + 25000
6. 78 inches
7. 17
8. $6.75
9. y = x + 1
10. y = 5x – 20