![Page 1: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/1.jpg)
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
![Page 2: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/2.jpg)
Pre-Algebra
11-2
Slope of a Line
![Page 3: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/3.jpg)
Learn to find the slope of a line and use slope to understand and draw graphs.
![Page 4: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/4.jpg)
You looked at slope on the coordinate plane in Lesson 5-5 (p. 244).
Remember!
![Page 5: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/5.jpg)
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
![Page 6: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/6.jpg)
![Page 7: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/7.jpg)
![Page 8: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/8.jpg)
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
![Page 9: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/9.jpg)
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
![Page 10: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/10.jpg)
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
![Page 11: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/11.jpg)
Use the graph of the line to determine its slope.
Example: Finding Slope from a Graph
![Page 12: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/12.jpg)
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
![Page 13: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/13.jpg)
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
![Page 14: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/14.jpg)
Use the graph of the line to determine its slope.
Try This
![Page 15: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/15.jpg)
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
![Page 16: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/16.jpg)
Recall that two parallel lines have the same slope. The slopes of two perpendicular lines are negative reciprocals of each other.
![Page 17: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/17.jpg)
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
![Page 18: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/18.jpg)
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
![Page 19: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/19.jpg)
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
![Page 20: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/20.jpg)
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
![Page 21: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/21.jpg)
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
![Page 22: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/22.jpg)
1
2(3, 1)
Example Continued
![Page 23: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/23.jpg)
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
![Page 24: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/24.jpg)
1
2(1, 1)
Try This Continued
![Page 25: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/25.jpg)
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
![Page 26: 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](https://reader033.vdocuments.us/reader033/viewer/2022061603/56649e665503460f94b611c0/html5/thumbnails/26.jpg)
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