Pre-Algebra 11-2 Slope of a Line 11-2 Slope of a Line Pre-Algebra Warm Up Warm Up Problem of the Day...

Pre-Algebra

11-2 Slope of a Line11-2 Slope of a Line

Pre-Algebra

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Pre-Algebra

11-2 Slope of a Line

Warm UpEvaluate 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

Pre-Algebra


–8, –2, 4

Pre-Algebra


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

Pre-Algebra


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

Remember!

Pre-Algebra


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

Pre-Algebra


Pre-Algebra


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

Pre-Algebra


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

Additional Example 1: Finding Slope, Given Two Points

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=

Pre-Algebra


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

Try This: Example 1

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=

Pre-Algebra


Use the graph of the line to determine its slope.

Additional Example 2: Finding Slope from a Graph

Pre-Algebra


Additional Example 2 Continued

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

Pre-Algebra


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= –

Pre-Algebra


Use the graph of the line to determine its slope.

Try This: Example 2

Pre-Algebra


Try This: Example 2 Continued

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

Pre-Algebra


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

Pre-Algebra


Additional Example 3A: Identifying Parallel and Perpendicular Lines by Slope

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= –

Pre-Algebra


Additional Example 3B: 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)

Pre-Algebra


Try This: Example 3A


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= –

Pre-Algebra


Try This: Example 3B

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

Pre-Algebra


Additional Example 4: Graphing a Line Using a Point and the Slope

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

Pre-Algebra



1

2(3, 1)

Pre-Algebra


Try This: Example 4

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

Pre-Algebra


Try This: Example 4 Continued

1

2(1, 1)

Pre-Algebra


Lesson Quiz: Part 1

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–

Pre-Algebra


Lesson Quiz: Part 2


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

Pre-Algebra 11-2 Slope of a Line 11-2 Slope of a Line Pre-Algebra Warm Up Warm Up Problem of the Day...

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