Evaluate each equation for x = –1, 0, and 1.

1. y = 3x

2. y = x – 7

3. y = 2x + 5

4. y = 6x – 2

–3, 0, 3

–8, –7, –6

3, 5, 7

–8, –2, 4

Pre-Class Warm Up

Pre-Algebra

11-2

Slope of a Line

Learn to find the slope of a line and use slope to understand and draw graphs.

You looked at slope on the coordinate plane in Lesson 5-5 (p. 244).

Remember!

Linear equations have constant slope. For a line on the coordinate plane, slope is the following ratio:

vertical change horizontal change

change in y change in x=

This ratio is often referred to as , or “rise

over run,” where rise indicates the number of units moved up or down and run indicates the number of units moved to the left or right. Slope can be positive, negative, zero, or undefined. A line with positive slope goes up from left to right. A line with negative slope goes down from left to right.

rise run

If you know any two points on a line, or two solutions of a linear equation, you can find the slope of the line without graphing. The slope of a line through the points (x1, y1) and (x2, y2) is as follows:

yy22 –– yy11 xx22 –– xx11

Find the slope of the line that passes through (–2, –3) and (4, 6).

Let (x1, y1) be (–2, –3) and (x2, y2) be (4, 6).

6 – (–3)4 – (–2)

Substitute 6 for y2, –3 for y1, 4 for x2, and –2 for x1.

96=

The slope of the line that passes through (–

2, –3) and (4, 6) is . 32

=y2 – y1

x2 – x1

32=

Example: Finding Slope, Given Two Points

Find the slope of the line that passes through (–4, –6) and (2, 3).

Let (x1, y1) be (–4, –6) and (x2, y2) be (2, 3).

3 – (–6)2 – (–4)

Substitute 3 for y2, –6 for y1, 2 for x2, and –4 for x1.

96=

The slope of the line that passes through (–

4, –6) and (2, 3) is . 32

=y2 – y1

x2 – x1

32=

Try This

Use the graph of the line to determine its slope.

Example: Finding Slope from a Graph

Choose two points on the line: (0, 1) and (3, –4).

Guess by looking at the graph:

riserun = –5

3 = – 5 3

Use the slope formula.

Let (3, –4) be (x1, y1) and (0, 1) be (x2, y2).

1 – (–4) 0 – 3=

y2 – y1

x2 – x1

5–3= 5

3= –

–5

3

Example Continued

Notice that if you switch (x1, y1) and (x2, y2), you get the same slope:

53The slope of the given line is – .

Let (0, 1) be (x1, y1) and (3, –4) be (x2, y2).

–4 – 1 3 – 0=

y2 – y1

x2 – x1

–5 3= 5

3= –

Example Continued

Use the graph of the line to determine its slope.

Try This

Choose two points on the line: (1, 1) and (0, –1).

Guess by looking at the graph:

riserun = 2

1 = 2

Use the slope formula.

Let (1, 1) be (x1, y1) and (0, –1) be (x2, y2).

=y2 – y1

x2 – x1

–2–1=

–1 – 1 0 – 1

= 2

12

Try This Continued

Recall that two parallel lines have the same slope. The slopes of two perpendicular lines are negative reciprocals of each other.

Tell whether the lines passing through the given points are parallel or perpendicular.

A. line 1: (–6, 4) and (2, –5); line 2: (–1, –4) and (8, 4)

slope of line 1:

slope of line 2:

Line 1 has a slope equal to – and line 2 has a slope

equal to , – and are negative reciprocals of each

other, so the lines are perpendicular.

98

89

89

98

=y2 – y1

x2 – x1

–9 8= –5 – 4

2 – (–6)

4 – (–4)8 – (–1)=

y2 – y1

x2 – x1

8 9=

9 8= –

Example: Identifying Parallel and Perpendicular Lines by Slope

B. line 1: (0, 5) and (6, –2); line 2: (–1, 3) and (5, –4)

Both lines have a slope equal to – , so the lines are parallel.

76

slope of line 1:

slope of line 2:

=y2 – y1

x2 – x1

–7 6= –2 – 5

6 – 0

=y2 – y1

x2 – x1

7 6= –

–7 6= 7

6= – –4 – 35 – (–1)

Example: Identifying Parallel and Perpendicular Lines by Slope

Tell whether the lines passing through the given points are parallel or perpendicular.

A. line 1: (–8, 2) and (0, –7); line 2: (–3, –6) and (6, 2)

slope of line 1:

slope of line 2:

Line 1 has a slope equal to – and line 2 has a slope

equal to , – and are negative reciprocals of each

other, so the lines are perpendicular.

98

89

89

98

=y2 – y1

x2 – x1

–9 8= –7 – 2

0 – (–8)

2 – (–6)6 – (–3)=

y2 – y1

x2 – x1

8 9=

9 8= –

Try This

B. line 1: (1, 1) and (2, 2); line 2: (1, –2) and (2, -1)

Line 1 has a slope equal to 1 and line 2 has a slope equal to –1. 1 and –1 are negative reciprocals of each other, so the lines are perpendicular.

slope of line 1:

slope of line 2:

=y2 – y1

x2 – x1

1 1= 2 – 1

2 – 1

=y2 – y1

x2 – x1

–1 1= –1 – (–2)

2 – (1)

= 1

= –1

Try This

Graph the line passing through (3, 1) with slope 2.

Plot the point (3, 1). Then move 2 units up and right 1 unit and plot the point (4, 3). Use a straightedge to connect the two points.

The slope is 2, or . So for every 2 units up, you will move right 1 unit, and for every 2 units down, you will move left 1 unit.

21

Example: Graphing a Line Using a Point and the Slope

1

2(3, 1)

Example Continued

Graph the line passing through (1, 1) with slope 2.

Plot the point (1, 1). Then move 2 units up and right 1 unit and plot the point (2, 3). Use a straightedge to connect the two points.

The slope is 2, or . So for every 2 units up, you will move right 1 unit, and for every 2 units down, you will move left 1 unit.

21

Try This

1

2(1, 1)

Try This Continued

Find the slope of the line passing through each pair of points.

1. (4, 3) and (–1, 1)

2. (–1, 5) and (4, 2)

3. Use the graph of the line to

determine its slope.

25

53–

34–

Lesson Quiz: Part 1

Tell whether the lines passing through the given points are parallel or perpendicular.

4. line 1: (–2, 1), (2, –1); line 2: (0, 0), (–1, –2)

5. line 1: (–3, 1), (–2, 3); line 2: (2, 1), (0, –3)

parallel

perpendicular

Lesson Quiz: Part 2