58 slopes of lines
TRANSCRIPT
About Slopes
Definition of Slope About Slopes
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line,
(x1, y1)
(x2, y2)
About Slopes
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
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
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
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
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
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
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
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
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
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
Example A. Find the slope of each of the following lines. About Slopes
Example A. Find the slope of each of the following lines.
Two points are(–3, 1), (4, 1).
About Slopes
Example A. Find the slope of each of the following lines.
Two points are(–3, 1), (4, 1).Δy = 1 – (1) = 0
About Slopes
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
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
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
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
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
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
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
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
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
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
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
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)
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)
Lines that go through the quadrants I and III have positive slopes.
About Slopes
Lines that go through the quadrants I and III have positive slopes.
About Slopes
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.
About Slopes
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.
About Slopes
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.
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
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
Given a linear equation in x and y, solve for the variable y if possible, we get y = mx + b
About Slopes
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
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
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.
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.
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.
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.
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.
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.
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).
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.
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.
b. 0 = –2y + 6About Slopes
b. 0 = –2y + 6 solve for yAbout Slopes
b. 0 = –2y + 6 solve for y 2y = 6 y = 3
About Slopes
b. 0 = –2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3
About Slopes
b. 0 = –2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3
Hence the slope m is 0.
About Slopes
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
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
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
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
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.
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.
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.
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.
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.
Two Facts About SlopesI. Parallel lines have the same slope.
About Slopes
Two Facts About SlopesI. Parallel lines have the same slope.II. Slopes of perpendicular lines are the negative reciprocal of each other.
About Slopes
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
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
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
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
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
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
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?
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
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
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
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
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.
About Slopes
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
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
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