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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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 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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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 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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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).

12

–6 –4 –2 0 2 4 60

2

4

6y

x

–2

–4

–6

y = x + 212

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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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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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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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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

34

15

Solution follows…

15161718

–6 –4 –2 0 2 4 60

2

4

6y

x

–2

–4

–6

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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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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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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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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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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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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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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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Solution follows…


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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13

11121315

–6 –4 –2 0 2 4 60

2

4

6y

x

–2

–4

–6

14

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Solution follows…


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).

43

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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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Solution follows…


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.


1 topic 4.4.3 the slope-intercept form of a line the slope-intercept form of a line

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