Pre-Algebra

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

Pre-Algebra

Homework & Learning GoalHomework & Learning GoalLesson PresentationLesson Presentation

Pre-Algebra

11-2 Slope of a LineHOMEWORK answers

Page 543 #1-11

Pre-Algebra

11-2 Slope of a LinePre-Algebra HOMEWORK

Page 548#1-5

Pre-Algebra

11-2 Slope of a Line

Our Learning GoalStudents will be able to graph lines using linear equations,

understand the slope of a line and graph inequalities.

Pre-Algebra

11-2 Slope of a LineOur Learning Goal Assignments

• Learn to identify and graph linear equations.• Learn to find the slope of a line and use slope to understand

and draw graphs. • Learn to use slopes and intercepts to graph linear equations. • Learn to find the equation of a line given one point and the

slope. • Learn to recognize direct variation by graphing tables of data

and checking for constant ratios.• Learn to graph inequalities on the coordinate plane. • Learn to recognize relationships in data and find the equation

of a line of best fit.

Pre-Algebra

11-2 Slope of a Line

Today’s Learning Goal Assignment

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

Pre-Algebra

11-2 Slope of a Line

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

Remember!

Pre-Algebra

11-2 Slope of a Line

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

horizontal changechange 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

11-2 Slope of a Line

Pre-Algebra

11-2 Slope of a Line

Pre-Algebra

11-2 Slope of a Line

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

11-2 Slope of a Line

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 x1.9

6=

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

2

=y2 – y1x2 – x1

32=

Pre-Algebra

11-2 Slope of a Line

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 x1.9

6=

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

2

=y2 – y1x2 – x1

32=

Pre-Algebra

11-2 Slope of a Line

Use the graph of the line to determine its slope.

Additional Example 2: Finding Slope from a Graph

Pre-Algebra

11-2 Slope of a LineAdditional Example 2 Continued

Choose two points on the line: (0, 1) and (3, –4).Guess by looking at the graph:

riserun = –5

3 = – 5 3Use 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

11-2 Slope of a Line

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).

Additional Example 2 Continued

–4 – 1 3 – 0=y2 – y1

x2 – x1–5 3= 5

3= –

Pre-Algebra

11-2 Slope of a Line

Use the graph of the line to determine its slope.

Try This: Example 2

Pre-Algebra

11-2 Slope of a LineTry 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 – y1x2 – x1

–2–1=

–1 – 1 0 – 1

= 2

12

Pre-Algebra

11-2 Slope of a Line

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

Pre-Algebra

11-2 Slope of a LineAdditional Example 3A: Identifying Parallel and

Perpendicular Lines by SlopeTell 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 – y1x2 – x1

–9 8= –5 – 4

2 – (–6)4 – (–4)8 – (–1)=y2 – y1

x2 – x1 8 9=

9 8= –

Pre-Algebra

11-2 Slope of a LineAdditional 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 – y1x2 – x1

–7 6= –2 – 5

6 – 0

=y2 – y1x2 – x1

7 6= –

–7 6= 7

6= – –4 – 35 – (–1)

Pre-Algebra

11-2 Slope of a LineTry This: Example 3A

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 – y1x2 – x1

–9 8= –7 – 2

0 – (–8)2 – (–6)6 – (–3)=y2 – y1

x2 – x1 8 9=

9 8= –

Pre-Algebra

11-2 Slope of a Line

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 – y1x2 – x1

1 1= 2 – 1

2 – 1

=y2 – y1x2 – x1

–1 1= –1 – (–2)

2 – (1)

= 1

= –1

Pre-Algebra

11-2 Slope of a LineAdditional Example 4: Graphing a Line Using a Point and

the SlopeGraph 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

11-2 Slope of a Line

Additional Example 4 Continued

12

(3, 1)

Pre-Algebra

11-2 Slope of a Line

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

11-2 Slope of a Line

Try This: Example 4 Continued

12

(1, 1)

Pre-Algebra

11-2 Slope of a Line

Lesson Quiz: Part 1Find 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

11-2 Slope of a Line

Lesson Quiz: Part 2Tell 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