1 topic 4.4.3 the slope-intercept form of a line the slope-intercept form of a line
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Topic 4.4.3Topic 4.4.3
The Slope-Intercept Form of a Line
The Slope-Intercept Form of a Line
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The Slope-Intercept Form of a LineThe Slope-Intercept Form of a Line
California Standard: 8.0 Students understand the concepts of parallel lines and perpendicular lines and how their slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point.
What it means for you:You’ll plot graphs and solve equations using a method called the slope-intercept form of an equation.
Key Words:• slope-intercept form• intercept
Topic4.4.3
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The Slope-Intercept Form of a LineThe Slope-Intercept Form of a Line
Now that you’ve practiced calculating the slope and intercept of a line, you can use the two things together to make plotting graphs easier.
Topic4.4.3
–6 –4 –2 0 2 4 60
2
4
6y
x
–2
–4
–6
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The Slope-Intercept Form of a LineThe Slope-Intercept Form of a Line
The Slope-Intercept Form of a Line: y = mx + b
In the slope-intercept form the y is alone on one side of the equation.
Topic4.4.3
Here are a few examples of equations in the slope-intercept form:
The slope-intercept form of the equation is:
y = mx + b
y = 3x + 2 y = – x + 1 y = –x + 1 y = x – 423
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The Slope-Intercept Form of a LineThe Slope-Intercept Form of a Line
Guided Practice
In Exercises 1–6, decide whether each equation is in slope-intercept form or not.
Topic4.4.3
1. y = 3x + 7 2. 3x + 4y = 7
3. y – 3 = 2(x – 4) 4. y – 8 = 3(x – 4)
5. y = x + 18 6. y = –4x – 1
Solution follows…
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yes
noyes
no
yes
no
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The Slope-Intercept Form of a LineThe Slope-Intercept Form of a Line
The Slope-Intercept Form Makes It Easy to Plot Graphs
The slope-intercept form of an equation is really useful for plotting graphs because m is the slope of the line and b is the y-coordinate of the y-intercept.
Topic4.4.3
y = mx + by-coordinate of y-interceptslope
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The Slope-Intercept Form of a LineThe Slope-Intercept Form of a Line
Example 1
Plot the graph of y = x + 2.
Solution follows…
Solution
Topic4.4.3
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y = mx + b = x + 212
— for a slope of , go up
1 unit for every 2 units across.
Slope = m = 12
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y-coordinate of the y-intercept = b = 2— so the y-intercept is (0, 2).
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–6 –4 –2 0 2 4 60
2
4
6y
x
–2
–4
–6
y = x + 212
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The Slope-Intercept Form of a LineThe Slope-Intercept Form of a Line
Example 2
Plot the graph of y = –3x – 4.
Solution follows…
Solution
Topic4.4.3
y = mx + b = –3x – 4
Slope = m = –3— for a slope of –3, go down 3 units for every unit across.
y-coordinate of the y-intercept = b = –4— so the y-intercept is (0, –4).
–6 –4 –2 0 2 4 60
2
4
6y
x
–2
–4
–6
y = –3x – 4
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The Slope-Intercept Form of a LineThe Slope-Intercept Form of a Line
Guided Practice
7. In the equation y = 8x + 5, find the slope.
8. In the equation y = –x + 10, find the slope.
9. In the equation y = 2x + 5, find the y-intercept.
10. In the equation y = 7b – 3, find the y-intercept.
Topic4.4.3
Solution follows…
y-intercept = (0, 5)
m = 8
m = –1
y-intercept = (0, –3)
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The Slope-Intercept Form of a LineThe Slope-Intercept Form of a Line
Guided Practice
In Exercises 11–14, plot each equation on a graph.
Topic4.4.3
Solution follows…
11. y = 2x + 3
12. y = x – 6
13. y = –7x – 8
14. y = – x – 413
–6 –4 –2 0 2 4 60
2
4
6y
x
–2
–4
–6
11121314
–6 –4 –2 0 2 4 6
–2
0
2
4y
x
–4
–6
–8
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The Slope-Intercept Form of a LineThe Slope-Intercept Form of a Line
Guided Practice
In Exercises 15–18, plot each equation on a graph.
Topic4.4.3
15. y = – x + 6
16. y = x – 3
17. y = x
18. y = 6
12
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Solution follows…
15161718
–6 –4 –2 0 2 4 60
2
4
6y
x
–2
–4
–6
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The Slope-Intercept Form of a LineThe Slope-Intercept Form of a Line
Solve for y to Get the Slope-Intercept Form
OK, so you know the slope-intercept form of an equation makes drawing graphs a lot easier.
Topic4.4.3
To get the equation into slope-intercept form, solve for y.
The trouble is, you’ll often be given an equation which isn’t in slope-intercept form.
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The Slope-Intercept Form of a LineThe Slope-Intercept Form of a Line
Example 3
A line has the equation Ax + By = C, where B 0. Solve this equation for y, justifying each step.
Solution follows…
Solution
Topic4.4.3
Ax + By = C
Ax – Ax + By = –Ax + C
Given equation
Subtraction property of equality
By = –Ax + C
Solution continues…
Division property of equality
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The Slope-Intercept Form of a LineThe Slope-Intercept Form of a Line
Example 3
A line has the equation Ax + By = C, where B 0. Solve this equation for y, justifying each step.
Solution (continued)
Topic4.4.3
The equation is now in slope-intercept form.
The slope, m = –
and the y-coordinate of the y-intercept, b = – .
AB C
B
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The Slope-Intercept Form of a LineThe Slope-Intercept Form of a Line
Example 4
Determine the slope and y-intercept of the line 2x – 3y = 9.
Solution follows…
Solution
Topic4.4.3
Step 1: Solve the given equation for y.
2x – 3y = 9
–3y = –2x + 9
y = x – 323
y =
Now you’ve got the equation in slope-intercept form, y = mx + b. Solution continues…
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The Slope-Intercept Form of a LineThe Slope-Intercept Form of a Line
Example 4
Determine the slope and y-intercept of the line 2x – 3y = 9.
Solution (continued)
Topic4.4.3
Step 2: Get the slope and y-intercept from the equation.
The slope, m = . 23
The y-coordinate of the y-intercept, b = –3. So, the y-intercept = (0, –3).
y = x – 323
Step 1:
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The Slope-Intercept Form of a LineThe Slope-Intercept Form of a Line
Guided Practice
In Exercises 19-24, find the slope and y-intercept of the line.
Topic4.4.3
19. 3x + 3y = 9 20. 2y – 6x = 10
21. 2x – 2y = 5 22. –7y + 5x = 14
23. –2y + 3x = 8 24. –5x + 4y = 12
Solution follows…
m = –1, (0, 3)
x + y = 3 y = –x + 3
m = 3, (0, 5)
y – 3x = 5 y = 3x + 5
m = 1, (0, – )52
x – y = y = x –52
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m = , (0, –2)57
y – x = –2 y = x – 257
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m = , (0, –4)32
y – x = –4 y = x – 432
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m = , (0, 3)54
– x + y = 3 y = x + 354
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The Slope-Intercept Form of a LineThe Slope-Intercept Form of a Line
Guided Practice
Write down the equations of the following lines in slope-intercept form.
Topic4.4.3
25. The line with slope 4 that passes through the point (0, 2).
26. The line with slope 2 that passes through the point (0, –6).
27. The line with slope –3 that passes through the point (0, 1).
Solution follows…
y = 4x + 2
y = –3x + 1
y = 2x – 6
67
28. The line with slope – that passes through the point (0, –3).
67y = – x – 3
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Independent Practice
Solution follows…
The Slope-Intercept Form of a LineThe Slope-Intercept Form of a LineTopic4.4.3
In Exercises 1–10, find the slope and y-intercept of each equation that’s given.
3. y = 3x + 6 4. –6y = 3x + 12
5. y = 2(x + 2) 6. y – 4 = 2(x + 1)
7. 7 – y = 5(x + 4) 8. y – 3 = 2(x – 9)
9. 3x + 4y = 8 10. 2x + 3y = 9
1. y = x – 5 2. y = x + 112
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m = 3, (0, 6)
m = 2, (0, 4) m = 2, (0, 6)
m = –5, (0, –13) m = 2, (0, –15)
34m = – , (0, 2)
m = , (0, –5)12
m = – , (0, 3)23
m = , (0, 1)23
m = – , (0, –2)12
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Independent Practice
Solution follows…
The Slope-Intercept Form of a LineThe Slope-Intercept Form of a LineTopic4.4.3
In Exercises 11–15, plot the graph of the given equation.
11. y = x + 5
12. y = – x – 6
13. y = x + 2
14. y = –x + 2
15. y = 2x
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11121315
–6 –4 –2 0 2 4 60
2
4
6y
x
–2
–4
–6
14
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Independent Practice
Solution follows…
The Slope-Intercept Form of a LineThe Slope-Intercept Form of a LineTopic4.4.3
In Exercises 16–20, write the equations of the lines in slope-intercept form.
16. A line with slope that passes through the point (0, 4).
17. A line with slope that passes through the point (0, –2).
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18. 4x + 2y = 8 19. 6x – 3y = 15
20. 3x – 4y = –16
y = –2x + 4 y = 2x – 5
y = x + 434
y = x + 443
y = x – 212
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Independent Practice
Solution follows…
The Slope-Intercept Form of a LineThe Slope-Intercept Form of a LineTopic4.4.3
In Exercises 21–22, write the equations of the lines in slope-intercept form.
21. The line with slope 0 that passes through the point (2, 6)
22. The line with slope 2 that passes through the point (6, 3)
y = 6
y = 2x – 9
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Round UpRound Up
If m = 0, the slope-intercept form of the equation becomes y = b — the equation of a horizontal line.
If b = 0, the equation becomes y = mx, meaning the line passes through the origin.
The Slope-Intercept Form of a LineThe Slope-Intercept Form of a LineTopic4.4.3