37 more on slopes-x
TRANSCRIPT
Frank Ma © 2011
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Definition of Slope More on Slopes
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line,
(x1, y1)
(x2, y2)
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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)
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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)
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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)
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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.
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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.
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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”.
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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=
More on Slopes
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
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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
More on Slopes
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
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Example A. Find the slope of each of the following lines. More on Slopes
Example A. Find the slope of each of the following lines.
Two points are(–3, 1), (4, 1).
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Example A. Find the slope of each of the following lines.
Two points are(–3, 1), (4, 1).Δy = 1 – (1) = 0
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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
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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
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m = ΔyΔx = 0
7 = 0
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
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m = ΔyΔx = 0
7Horizontal line Slope = 0
= 0
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
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m = ΔyΔx = 0
7Horizontal line Slope = 0
= 0
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
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m = ΔyΔx = 0
7Horizontal line Slope = 0
= 0
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
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m = ΔyΔx = 0
7Horizontal line Slope = 0
= 0
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
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ΔyΔx = 7
4m = ΔyΔx = 0
7Horizontal line Slope = 0
= 0
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
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ΔyΔx = 7
4m = ΔyΔx = 0
7Horizontal line Slope = 0
Tilted line Slope = 0
= 0
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).
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ΔyΔx = 7
4m = ΔyΔx = 0
7Horizontal line Slope = 0
Tilted line Slope = 0
= 0
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
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ΔyΔx = 7
4m = ΔyΔx = 0
7Horizontal line Slope = 0
Tilted line Slope = 0
= 0
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
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ΔyΔx = 7
4m = ΔyΔx = 0
7Horizontal line Slope = 0
Tilted line Slope = 0
= 0
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
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ΔyΔx = 7
4m = ΔyΔx = 0
7m = Δy
Δx = 70
Horizontal line Slope = 0
Tilted line Slope = 0
= 0 (UDF)
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
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ΔyΔx = 7
4m = ΔyΔx = 0
7m = Δy
Δx = 70
Horizontal line Slope = 0
Vertical line Slope is UDF
Tilted line Slope = 0
= 0 (UDF)
Lines that go through the quadrants I and III have positive slopes.
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Lines that go through the quadrants I and III have positive slopes.
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III
III IV
Lines that go through the quadrants I and III have positive slopes.
Lines that go through the quadrants II and IV have negative slopes.
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III
III IV
Lines that go through the quadrants I and III have positive slopes.
Lines that go through the quadrants II and IV have negative slopes.
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III
III IV
III
III IV
Lines that go through the quadrants I and III have positive slopes.
Lines that go through the quadrants II and IV have negative slopes.
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The formula for slopes requires geometric information,i.e. the positions of two points on the line.
III
III IV
III
III IV
Lines that go through the quadrants I and III have positive slopes.
Lines that go through the quadrants II and IV have negative slopes.
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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
Given a linear equation in x and y, solve for the variable y if possible, we get y = mx + b
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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.
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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.
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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.
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a. 3x = –2y + 6
Example B. Write the equations into the slope intercept form, list the slopes, the y-intercepts and draw the lines.
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.
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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.
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.
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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.
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.
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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.
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.
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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.
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.
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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.
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.
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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).
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.
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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.
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.
More on 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.
b. 0 = –2y + 6More on Slopes
b. 0 = –2y + 6 solve for yMore on Slopes
b. 0 = –2y + 6 solve for y 2y = 6 y = 3
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b. 0 = –2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3
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b. 0 = –2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3
Hence the slope m is 0.
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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).
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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.
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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.
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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
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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
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The variable y can’t be isolated because there is no y.
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
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The variable y can’t be isolated because there is no y.Hence the slope is undefinedand this is a vertical line.
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
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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.
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
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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.
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
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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.
Two Facts About SlopesI. Parallel lines have the same slope.
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Two Facts About SlopesI. Parallel lines have the same slope.II. Slopes of perpendicular lines are the negative reciprocal of each other.
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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?
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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
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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
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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
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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.
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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.
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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.
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b. What is the slope of L if L is perpendicular to 3x = 2y + 4?
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.
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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
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.
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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
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.
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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
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.
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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
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=
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.
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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)
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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.
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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.
More on SlopesFind the slope, if possible of each of the following lines