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Pre-Algebra
11-2 Slope of a Line11-2 Slope of a Line
Pre-Algebra
Homework & Learning GoalHomework & Learning GoalLesson PresentationLesson Presentation
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Pre-Algebra
11-2 Slope of a LineHOMEWORK answers
Page 543 #1-11
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Pre-Algebra
11-2 Slope of a LinePre-Algebra HOMEWORK
Page 548#1-5
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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.
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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.
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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.
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Pre-Algebra
11-2 Slope of a Line
You looked at slope on the coordinate plane in Lesson 5-5 (p. 244).
Remember!
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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
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Pre-Algebra
11-2 Slope of a Line
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Pre-Algebra
11-2 Slope of a Line
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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
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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=
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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=
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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
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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
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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= –
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Pre-Algebra
11-2 Slope of a Line
Use the graph of the line to determine its slope.
Try This: Example 2
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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
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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.
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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= –
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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)
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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= –
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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
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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
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Pre-Algebra
11-2 Slope of a Line
Additional Example 4 Continued
12
(3, 1)
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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
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Pre-Algebra
11-2 Slope of a Line
Try This: Example 4 Continued
12
(1, 1)
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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–
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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