58 slopes of lines
TRANSCRIPT
![Page 1: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/1.jpg)
About Slopes
![Page 2: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/2.jpg)
Definition of Slope About Slopes
![Page 3: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/3.jpg)
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line,
(x1, y1)
(x2, y2)
About Slopes
![Page 4: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/4.jpg)
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔxm =
(x1, y1)
(x2, y2)
About Slopes
![Page 5: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/5.jpg)
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔx
y2 – y1
x2 – x1m = =
(x1, y1)
(x2, y2)
About Slopes
![Page 6: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/6.jpg)
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔx
y2 – y1
x2 – x1m = =
Geometry of Slope
(x1, y1)
(x2, y2)
About Slopes
![Page 7: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/7.jpg)
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔx
y2 – y1
x2 – x1m = =
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points.
About Slopes
![Page 8: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/8.jpg)
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔx
y2 – y1
x2 – x1m = =
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points.Δx = x2 – x1 = the difference in the runs of the points.
About Slopes
![Page 9: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/9.jpg)
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔx
y2 – y1
x2 – x1m = =
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points.Δx = x2 – x1 = the difference in the runs of the points.
ΔyΔx= Therefore m is the ratio of the “rise” to the
“run”.
About Slopes
![Page 10: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/10.jpg)
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔx
y2 – y1
x2 – x1m = =
riserun=
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points.Δx = x2 – x1 = the difference in the runs of the points.
ΔyΔx= Therefore m is the ratio of the “rise” to the
“run”. m = Δy
Δxy2 – y1
x2 – x1=
About Slopes
![Page 11: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/11.jpg)
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔx
y2 – y1
x2 – x1m = =
riserun=
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points.Δx = x2 – x1 = the difference in the runs of the points.
ΔyΔx= Therefore m is the ratio of the “rise” to the
“run”. m = Δy
Δxy2 – y1
x2 – x1=
easy to memorize
About Slopes
![Page 12: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/12.jpg)
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔx
y2 – y1
x2 – x1m = =
riserun=
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points.Δx = x2 – x1 = the difference in the runs of the points.
ΔyΔx= Therefore m is the ratio of the “rise” to the
“run”. m = Δy
Δxy2 – y1
x2 – x1=
easy to memorize
the exact formula
About Slopes
![Page 13: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/13.jpg)
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔx
y2 – y1
x2 – x1m = =
riserun=
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points.Δx = x2 – x1 = the difference in the runs of the points.
ΔyΔx= Therefore m is the ratio of the “rise” to the
“run”. m = Δy
Δxy2 – y1
x2 – x1=
easy to memorize
the exact formula
geometric meaning
About Slopes
![Page 14: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/14.jpg)
Example A. Find the slope of each of the following lines. About Slopes
![Page 15: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/15.jpg)
Example A. Find the slope of each of the following lines.
Two points are(–3, 1), (4, 1).
About Slopes
![Page 16: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/16.jpg)
Example A. Find the slope of each of the following lines.
Two points are(–3, 1), (4, 1).Δy = 1 – (1) = 0
About Slopes
![Page 17: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/17.jpg)
Example A. Find the slope of each of the following lines.
Two points are(–3, 1), (4, 1).Δy = 1 – (1) = 0Δx = 4 – (–3) = 7
About Slopes
![Page 18: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/18.jpg)
Example A. Find the slope of each of the following lines.
Two points are(–3, 1), (4, 1).Δy = 1 – (1) = 0Δx = 4 – (–3) = 7
About Slopes
m = ΔyΔx = 0
7 = 0
![Page 19: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/19.jpg)
Example A. Find the slope of each of the following lines.
Two points are(–3, 1), (4, 1).Δy = 1 – (1) = 0Δx = 4 – (–3) = 7
About Slopes
m = ΔyΔx = 0
7Horizontal line Slope = 0
= 0
![Page 20: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/20.jpg)
Example A. Find the slope of each of the following lines.
Two points are(–2, –4), (2, 3).
Two points are(–3, 1), (4, 1).Δy = 1 – (1) = 0Δx = 4 – (–3) = 7
About Slopes
m = ΔyΔx = 0
7Horizontal line Slope = 0
= 0
![Page 21: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/21.jpg)
Example A. Find the slope of each of the following lines.
Two points are(–2, –4), (2, 3).Δy = 3 – (–4) = 7
Two points are(–3, 1), (4, 1).Δy = 1 – (1) = 0Δx = 4 – (–3) = 7
About Slopes
m = ΔyΔx = 0
7Horizontal line Slope = 0
= 0
![Page 22: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/22.jpg)
Example A. Find the slope of each of the following lines.
Two points are(–2, –4), (2, 3).Δy = 3 – (–4) = 7Δx = 2 – (–2) = 4
Two points are(–3, 1), (4, 1).Δy = 1 – (1) = 0Δx = 4 – (–3) = 7
About Slopes
m = ΔyΔx = 0
7Horizontal line Slope = 0
= 0
![Page 23: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/23.jpg)
Example A. Find the slope of each of the following lines.
Two points are(–2, –4), (2, 3).Δy = 3 – (–4) = 7Δx = 2 – (–2) = 4
m =
Two points are(–3, 1), (4, 1).Δy = 1 – (1) = 0Δx = 4 – (–3) = 7
About Slopes
ΔyΔx = 7
4m = ΔyΔx = 0
7Horizontal line Slope = 0
= 0
![Page 24: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/24.jpg)
Example A. Find the slope of each of the following lines.
Two points are(–2, –4), (2, 3).Δy = 3 – (–4) = 7Δx = 2 – (–2) = 4
m =
Two points are(–3, 1), (4, 1).Δy = 1 – (1) = 0Δx = 4 – (–3) = 7
About Slopes
ΔyΔx = 7
4m = ΔyΔx = 0
7Horizontal line Slope = 0
Tilted line Slope = 0
= 0
![Page 25: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/25.jpg)
Example A. Find the slope of each of the following lines.
Two points are(–2, –4), (2, 3).Δy = 3 – (–4) = 7Δx = 2 – (–2) = 4
m =
Two points are(–3, 1), (4, 1).Δy = 1 – (1) = 0Δx = 4 – (–3) = 7
Two points are(–1, 3), (6, 3).
About Slopes
ΔyΔx = 7
4m = ΔyΔx = 0
7Horizontal line Slope = 0
Tilted line Slope = 0
= 0
![Page 26: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/26.jpg)
Example A. Find the slope of each of the following lines.
Two points are(–2, –4), (2, 3).Δy = 3 – (–4) = 7Δx = 2 – (–2) = 4
m =
Two points are(–3, 1), (4, 1).Δy = 1 – (1) = 0Δx = 4 – (–3) = 7
Two points are(–1, 3), (6, 3).Δy = 3 – 3 = 0
About Slopes
ΔyΔx = 7
4m = ΔyΔx = 0
7Horizontal line Slope = 0
Tilted line Slope = 0
= 0
![Page 27: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/27.jpg)
Example A. Find the slope of each of the following lines.
Two points are(–2, –4), (2, 3).Δy = 3 – (–4) = 7Δx = 2 – (–2) = 4
m =
Two points are(–3, 1), (4, 1).Δy = 1 – (1) = 0Δx = 4 – (–3) = 7
Two points are(–1, 3), (6, 3).Δy = 3 – 3 = 0Δx = 6 – (–1) = 7
About Slopes
ΔyΔx = 7
4m = ΔyΔx = 0
7Horizontal line Slope = 0
Tilted line Slope = 0
= 0
![Page 28: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/28.jpg)
Example A. Find the slope of each of the following lines.
Two points are(–2, –4), (2, 3).Δy = 3 – (–4) = 7Δx = 2 – (–2) = 4
m =
Two points are(–3, 1), (4, 1).Δy = 1 – (1) = 0Δx = 4 – (–3) = 7
Two points are(–1, 3), (6, 3).Δy = 3 – 3 = 0Δx = 6 – (–1) = 7
About Slopes
ΔyΔx = 7
4m = ΔyΔx = 0
7m = Δy
Δx = 70
Horizontal line Slope = 0
Tilted line Slope = 0
= 0 (UDF)
![Page 29: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/29.jpg)
Example A. Find the slope of each of the following lines.
Two points are(–2, –4), (2, 3).Δy = 3 – (–4) = 7Δx = 2 – (–2) = 4
m =
Two points are(–3, 1), (4, 1).Δy = 1 – (1) = 0Δx = 4 – (–3) = 7
Two points are(–1, 3), (6, 3).Δy = 3 – 3 = 0Δx = 6 – (–1) = 7
About Slopes
ΔyΔx = 7
4m = ΔyΔx = 0
7m = Δy
Δx = 70
Horizontal line Slope = 0
Vertical line Slope is UDF
Tilted line Slope = 0
= 0 (UDF)
![Page 30: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/30.jpg)
Lines that go through the quadrants I and III have positive slopes.
About Slopes
![Page 31: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/31.jpg)
Lines that go through the quadrants I and III have positive slopes.
About Slopes
III
III IV
![Page 32: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/32.jpg)
Lines that go through the quadrants I and III have positive slopes.
Lines that go through the quadrants II and IV have negative slopes.
About Slopes
III
III IV
![Page 33: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/33.jpg)
Lines that go through the quadrants I and III have positive slopes.
Lines that go through the quadrants II and IV have negative slopes.
About Slopes
III
III IV
III
III IV
![Page 34: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/34.jpg)
Lines that go through the quadrants I and III have positive slopes.
Lines that go through the quadrants II and IV have negative slopes.
About Slopes
The formula for slopes requires geometric information,i.e. the positions of two points on the line.
III
III IV
III
III IV
![Page 35: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/35.jpg)
Lines that go through the quadrants I and III have positive slopes.
Lines that go through the quadrants II and IV have negative slopes.
About Slopes
The formula for slopes requires geometric information,i.e. the positions of two points on the line. However, if a line is given by its equation instead, we may determine the slope from the equation directly.
III
III IV
III
III IV
![Page 36: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/36.jpg)
Given a linear equation in x and y, solve for the variable y if possible, we get y = mx + b
About Slopes
![Page 37: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/37.jpg)
Given a linear equation in x and y, solve for the variable y if possible, we get y = mx + bthe number m is the slope and b is the y-intercept.
About Slopes
![Page 38: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/38.jpg)
Given a linear equation in x and y, solve for the variable y if possible, we get y = mx + bthe number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be doneonly if the y-term is present.
About Slopes
![Page 39: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/39.jpg)
Given a linear equation in x and y, solve for the variable y if possible, we get y = mx + bthe number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be doneonly if the y-term is present.
About Slopes
a. 3x = –2y + 6
Example B. Write the equations into the slope intercept form, list the slopes, the y-intercepts and draw the lines.
![Page 40: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/40.jpg)
Given a linear equation in x and y, solve for the variable y if possible, we get y = mx + bthe number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be doneonly if the y-term is present.
About Slopes
a. 3x = –2y + 6 solve for y
Example B. Write the equations into the slope intercept form, list the slopes, the y-intercepts and draw the lines.
![Page 41: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/41.jpg)
Given a linear equation in x and y, solve for the variable y if possible, we get y = mx + bthe number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be doneonly if the y-term is present.
About Slopes
a. 3x = –2y + 6 solve for y 2y = –3x + 6
Example B. Write the equations into the slope intercept form, list the slopes, the y-intercepts and draw the lines.
![Page 42: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/42.jpg)
Given a linear equation in x and y, solve for the variable y if possible, we get y = mx + bthe number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be doneonly if the y-term is present.
About Slopes
a. 3x = –2y + 6 solve for y 2y = –3x + 6
y = 2–3 x + 3
Example B. Write the equations into the slope intercept form, list the slopes, the y-intercepts and draw the lines.
![Page 43: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/43.jpg)
Given a linear equation in x and y, solve for the variable y if possible, we get y = mx + bthe number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be doneonly if the y-term is present.
About Slopes
a. 3x = –2y + 6 solve for y 2y = –3x + 6
y = 2–3 x + 3
Hence the slope m is –3/2
Example B. Write the equations into the slope intercept form, list the slopes, the y-intercepts and draw the lines.
![Page 44: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/44.jpg)
Given a linear equation in x and y, solve for the variable y if possible, we get y = mx + bthe number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be doneonly if the y-term is present.
About Slopes
a. 3x = –2y + 6 solve for y 2y = –3x + 6
y = 2–3 x + 3
Hence the slope m is –3/2 and the y-intercept is (0, 3).
Example B. Write the equations into the slope intercept form, list the slopes, the y-intercepts and draw the lines.
![Page 45: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/45.jpg)
Given a linear equation in x and y, solve for the variable y if possible, we get y = mx + bthe number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be doneonly if the y-term is present.
About Slopes
Example B. Write the equations into the slope intercept form, list the slopes, the y-intercepts and draw the lines.a. 3x = –2y + 6 solve for y 2y = –3x + 6
y = 2–3 x + 3
Hence the slope m is –3/2 and the y-intercept is (0, 3).Set y = 0, we get the x-intercept (2, 0).
![Page 46: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/46.jpg)
Given a linear equation in x and y, solve for the variable y if possible, we get y = mx + bthe number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be doneonly if the y-term is present.
About Slopes
a. 3x = –2y + 6 solve for y 2y = –3x + 6
y = 2–3 x + 3
Hence the slope m is –3/2 and the y-intercept is (0, 3).Set y = 0, we get the x-intercept (2, 0). Use these points to draw the line.
Example B. Write the equations into the slope intercept form, list the slopes, the y-intercepts and draw the lines.
![Page 47: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/47.jpg)
Given a linear equation in x and y, solve for the variable y if possible, we get y = mx + bthe number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be doneonly if the y-term is present.
About Slopes
a. 3x = –2y + 6 solve for y 2y = –3x + 6
y = 2–3 x + 3
Hence the slope m is –3/2 and the y-intercept is (0, 3).Set y = 0, we get the x-intercept (2, 0). Use these points to draw the line.
Example B. Write the equations into the slope intercept form, list the slopes, the y-intercepts and draw the lines.
![Page 48: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/48.jpg)
b. 0 = –2y + 6About Slopes
![Page 49: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/49.jpg)
b. 0 = –2y + 6 solve for yAbout Slopes
![Page 50: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/50.jpg)
b. 0 = –2y + 6 solve for y 2y = 6 y = 3
About Slopes
![Page 51: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/51.jpg)
b. 0 = –2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3
About Slopes
![Page 52: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/52.jpg)
b. 0 = –2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3
Hence the slope m is 0.
About Slopes
![Page 53: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/53.jpg)
b. 0 = –2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3
Hence the slope m is 0. The y-intercept is (0, 3).
About Slopes
![Page 54: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/54.jpg)
b. 0 = –2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3
Hence the slope m is 0. The y-intercept is (0, 3). There is no x-intercept.
About Slopes
![Page 55: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/55.jpg)
b. 0 = –2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3
Hence the slope m is 0. The y-intercept is (0, 3). There is no x-intercept.
About Slopes
![Page 56: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/56.jpg)
b. 0 = –2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3
Hence the slope m is 0. The y-intercept is (0, 3). There is no x-intercept.
c. 3x = 6
About Slopes
![Page 57: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/57.jpg)
b. 0 = –2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3
Hence the slope m is 0. The y-intercept is (0, 3). There is no x-intercept.
c. 3x = 6
About Slopes
The variable y can’t be isolated because there is no y.
![Page 58: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/58.jpg)
b. 0 = –2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3
Hence the slope m is 0. The y-intercept is (0, 3). There is no x-intercept.
c. 3x = 6
About Slopes
The variable y can’t be isolated because there is no y.Hence the slope is undefinedand this is a vertical line.
![Page 59: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/59.jpg)
b. 0 = –2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3
Hence the slope m is 0. The y-intercept is (0, 3). There is no x-intercept.
c. 3x = 6
About Slopes
The variable y can’t be isolated because there is no y.Hence the slope is undefinedand this is a vertical line.Solve for x 3x = 6 x = 2.
![Page 60: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/60.jpg)
b. 0 = –2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3
Hence the slope m is 0. The y-intercept is (0, 3). There is no x-intercept.
c. 3x = 6
About Slopes
The variable y can’t be isolated because there is no y.Hence the slope is undefinedand this is a vertical line.Solve for x 3x = 6 x = 2.This is the vertical line x = 2.
![Page 61: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/61.jpg)
b. 0 = –2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3
Hence the slope m is 0. The y-intercept is (0, 3). There is no x-intercept.
c. 3x = 6
About Slopes
The variable y can’t be isolated because there is no y.Hence the slope is undefinedand this is a vertical line.Solve for x 3x = 6 x = 2.This is the vertical line x = 2.
![Page 62: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/62.jpg)
Two Facts About SlopesI. Parallel lines have the same slope.
About Slopes
![Page 63: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/63.jpg)
Two Facts About SlopesI. Parallel lines have the same slope.II. Slopes of perpendicular lines are the negative reciprocal of each other.
About Slopes
![Page 64: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/64.jpg)
Two Facts About SlopesI. Parallel lines have the same slope.II. Slopes of perpendicular lines are the negative reciprocal of each other.Example C. a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
About Slopes
![Page 65: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/65.jpg)
Two Facts About SlopesI. Parallel lines have the same slope.II. Slopes of perpendicular lines are the negative reciprocal of each other.Example C. a. The line L is parallel to 4x – 2y = 5, what is the slope of L? Solve for y for 4x – 2y = 5
About Slopes
![Page 66: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/66.jpg)
Two Facts About SlopesI. Parallel lines have the same slope.II. Slopes of perpendicular lines are the negative reciprocal of each other.Example C. a. The line L is parallel to 4x – 2y = 5, what is the slope of L? Solve for y for 4x – 2y = 5 4x – 5 = 2y
About Slopes
![Page 67: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/67.jpg)
Two Facts About SlopesI. Parallel lines have the same slope.II. Slopes of perpendicular lines are the negative reciprocal of each other.Example C. a. The line L is parallel to 4x – 2y = 5, what is the slope of L? Solve for y for 4x – 2y = 5 4x – 5 = 2y 2x – 5/2 = y
About Slopes
![Page 68: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/68.jpg)
Two Facts About SlopesI. Parallel lines have the same slope.II. Slopes of perpendicular lines are the negative reciprocal of each other.Example C. a. The line L is parallel to 4x – 2y = 5, what is the slope of L? Solve for y for 4x – 2y = 5 4x – 5 = 2y 2x – 5/2 = y So the slope of 4x – 2y = 5 is 2.
About Slopes
![Page 69: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/69.jpg)
Two Facts About SlopesI. Parallel lines have the same slope.II. Slopes of perpendicular lines are the negative reciprocal of each other.Example C. a. The line L is parallel to 4x – 2y = 5, what is the slope of L? Solve for y for 4x – 2y = 5 4x – 5 = 2y 2x – 5/2 = y So the slope of 4x – 2y = 5 is 2. Since L is parallel to it , so L has slope 2 also.
About Slopes
![Page 70: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/70.jpg)
Two Facts About SlopesI. Parallel lines have the same slope.II. Slopes of perpendicular lines are the negative reciprocal of each other.Example C. a. The line L is parallel to 4x – 2y = 5, what is the slope of L? Solve for y for 4x – 2y = 5 4x – 5 = 2y 2x – 5/2 = y So the slope of 4x – 2y = 5 is 2. Since L is parallel to it , so L has slope 2 also.
About Slopes
b. What is the slope of L if L is perpendicular to 3x = 2y + 4?
![Page 71: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/71.jpg)
Two Facts About SlopesI. Parallel lines have the same slope.II. Slopes of perpendicular lines are the negative reciprocal of each other.Example C. a. The line L is parallel to 4x – 2y = 5, what is the slope of L? Solve for y for 4x – 2y = 5 4x – 5 = 2y 2x – 5/2 = y So the slope of 4x – 2y = 5 is 2. Since L is parallel to it , so L has slope 2 also.
About Slopes
b. What is the slope of L if L is perpendicular to 3x = 2y + 4? Solve for y to find the slope of 3x – 4 = 2y
![Page 72: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/72.jpg)
Two Facts About SlopesI. Parallel lines have the same slope.II. Slopes of perpendicular lines are the negative reciprocal of each other.Example C. a. The line L is parallel to 4x – 2y = 5, what is the slope of L? Solve for y for 4x – 2y = 5 4x – 5 = 2y 2x – 5/2 = y So the slope of 4x – 2y = 5 is 2. Since L is parallel to it , so L has slope 2 also.
About Slopes
b. What is the slope of L if L is perpendicular to 3x = 2y + 4? Solve for y to find the slope of 3x – 4 = 2y x – 2 = y 2
3
![Page 73: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/73.jpg)
Two Facts About SlopesI. Parallel lines have the same slope.II. Slopes of perpendicular lines are the negative reciprocal of each other.Example C. a. The line L is parallel to 4x – 2y = 5, what is the slope of L? Solve for y for 4x – 2y = 5 4x – 5 = 2y 2x – 5/2 = y So the slope of 4x – 2y = 5 is 2. Since L is parallel to it , so L has slope 2 also.
About Slopes
b. What is the slope of L if L is perpendicular to 3x = 2y + 4? Solve for y to find the slope of 3x – 4 = 2y x – 2 = y Hence the slope of 3x = 2y + 4 is .
2 3
2 3
![Page 74: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/74.jpg)
Two Facts About SlopesI. Parallel lines have the same slope.II. Slopes of perpendicular lines are the negative reciprocal of each other.Example C. a. The line L is parallel to 4x – 2y = 5, what is the slope of L? Solve for y for 4x – 2y = 5 4x – 5 = 2y 2x – 5/2 = y So the slope of 4x – 2y = 5 is 2. Since L is parallel to it , so L has slope 2 also.
About Slopes
b. What is the slope of L if L is perpendicular to 3x = 2y + 4? Solve for y to find the slope of 3x – 4 = 2y x – 2 = y Hence the slope of 3x = 2y + 4 is . So L has slope –2/3 since L is perpendicular to it.
2 3
2 3
![Page 75: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/75.jpg)
Summary on Slopes
How to Find SlopesI. If two points on the line are given, use the slope formula
II. If the equation of the line is given, solve for the y and get slope intercept form y = mx + b, then the number m is the slope.
Geometry of Slope The slope of tilted lines are nonzero. Lines with positive slopes connect quadrants I and III.Lines with negative slopes connect quadrants II and IV. Lines that have slopes with large absolute values are steep.The slope of a horizontal line is 0.A vertical lines does not have slope or that it’s UDF.Parallel lines have the same slopes.Perpendicular lines have the negative reciprocal slopes of each other.
riserun= m = Δy
Δxy2 – y1
x2 – x1=
![Page 76: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/76.jpg)
Exercise A. Identify the vertical and the horizontal lines by inspection first. Find their slopes or if it’s undefined, state so. Fine the slopes of the other ones by solving for the y.1. x – y = 3 2. 2x = 6 3. –y – 7= 0
4. 0 = 8 – 2x 5. y = –x + 4 6. 2x/3 – 3 = 6/5
7. 2x = 6 – 2y 8. 4y/5 – 12 = 3x/4 9. 2x + 3y = 3
10. –6 = 3x – 2y 11. 3x + 2 = 4y + 3x 12. 5x/4 + 2y/3 = 2 Exercise B. 13–18. Select two points and estimate the slope of each line.
13. 14. 15.
About Slopes
![Page 77: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/77.jpg)
16. 17. 18.Exercise C. Draw and find the slope of the line that passes through the given two points. Identify the vertical line and the horizontal lines by inspection first.19. (0, –1), (–2, 1) 20. (1, –2), (–2, 0) 21. (1, –2), (–2, –1)22. (3, –1), (3, 1) 23. (1, –2), (–2, 3) 24. (2, –1), (3, –1)25. (4, –2), (–3, 1) 26. (4, –2), (4, 0) 27. (7, –2), (–2, –6)28. (3/2, –1), (3/2, 1) 29. (3/2, –1), (1, –3/2)30. (–5/2, –1/2), (1/2, 1) 31. (3/2, 1/3), (1/3, 1/3)32. (–2/3, –1/4), (1/2, 2/3) 33. (3/4, –1/3), (1/3, 3/2)
About Slopes
![Page 78: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/78.jpg)
Exercise D. 34. Identify which lines are parallel and which one are perpendicular. A. The line that passes through (0, 1), (1, –2)
D. 2x – 4y = 1
B. C.
E. The line that’s perpendicular to 3y = xF. The line with the x–intercept at 3 and y intercept at 6. Find the slope, if possible of each of the following lines.35. The line passes with the x intercept at x = 2, and y–intercept at y = –5.
About Slopes
![Page 79: 58 slopes of lines](https://reader036.vdocuments.site/reader036/viewer/2022062412/58ed584b1a28ab4a1d8b45df/html5/thumbnails/79.jpg)
36. The equation of the line is 3x = –5y+737. The equation of the line is 0 = –5y+7 38. The equation of the line is 3x = 739. The line is parallel to 2y = 5 – 6x 40. the line is perpendicular to 2y = 5 – 6x41. The line is parallel to the line in problem 30. 42. the line is perpendicular to line in problem 31.43. The line is parallel to the line in problem 33. 44. the line is perpendicular to line in problem 34.
About SlopesFind the slope, if possible of each of the following lines