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37 more on slopes-x

The document defines slope as the ratio of the rise (change in y-values) to the run (change in x-values) between two points on a line. It provides the exact formula for calculating slope and gives examples of finding the slopes of various lines. Key points made include that horizontal lines have a slope of 0, vertical lines have an undefined slope, and lines through quadrants I and III have positive slopes while lines through quadrants II and IV have negative slopes.

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Frank Ma © 2011 More on Slopes
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) 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 Δy Δxm = (x1, y1) (x2, y2) 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 Δy Δx y2 – y1 x2 – x1 m = = (x1, y1) (x2, y2) 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 Δy Δx y2 – y1 x2 – x1 m = = Geometry of Slope (x1, y1) (x2, y2) 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 Δy Δx y2 – y1 x2 – x1 m = = (x1, y1) (x2, y2) Δy=y2–y1=rise Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points. 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 Δy Δx y2 – y1 x2 – x1 m = = (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. 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 Δy Δx y2 – y1 x2 – x1 m = = (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. Δy Δx =Therefore m is the ratio of the “rise” to the “run”. 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 Δy Δx y2 – y1 x2 – x1 m = = rise run= (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. Δy Δx =Therefore m is the ratio of the “rise” to the “run”. m = Δy Δx y2 – 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 Δy Δx y2 – y1 x2 – x1 m = = rise run= (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. Δy Δx =Therefore m is the ratio of the “rise” to the “run”. m = Δy Δx y2 – y1 x2 – x1 = easy to memorize 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 Δy Δx y2 – y1 x2 – x1 m = = rise run= (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. Δy Δx =Therefore m is the ratio of the “rise” to the “run”. m = Δy Δx y2 – 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 Δy Δx y2 – y1 x2 – x1 m = = rise run= (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. Δy Δx =Therefore m is the ratio of the “rise” to the “run”. m = Δy Δx y2 – y1 x2 – x1 = easy to memorize the exact formula geometric meaning More on Slopes
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). More on Slopes
Example A. Find the slope of each of the following lines. Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 More on Slopes
Example A. Find the slope of each of the following lines. Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 More on Slopes
Example A. Find the slope of each of the following lines. Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 More on Slopes m = Δy Δx = 0 7 = 0
Example A. Find the slope of each of the following lines. Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 More on Slopes m = Δy Δx = 0 7 Horizontal 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) = 0 Δx = 4 – (–3) = 7 More on Slopes m = Δy Δx = 0 7 Horizontal 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) = 0 Δx = 4 – (–3) = 7 More on Slopes m = Δy Δx = 0 7 Horizontal 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 Δx = 2 – (–2) = 4 Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 More on Slopes m = Δy Δx = 0 7 Horizontal 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 Δx = 2 – (–2) = 4 m = Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 More on Slopes Δy Δx = 7 4 m = Δy Δx = 0 7 Horizontal 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 Δx = 2 – (–2) = 4 m = Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 More on Slopes Δy Δx = 7 4 m = Δy Δx = 0 7 Horizontal 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) = 7 Δx = 2 – (–2) = 4 m = Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 Two points are (–1, 3), (6, 3). More on Slopes Δy Δx = 7 4 m = Δy Δx = 0 7 Horizontal 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) = 7 Δx = 2 – (–2) = 4 m = Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 Two points are (–1, 3), (6, 3). Δy = 3 – 3 = 0 More on Slopes Δy Δx = 7 4 m = Δy Δx = 0 7 Horizontal 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) = 7 Δx = 2 – (–2) = 4 m = Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 Two points are (–1, 3), (6, 3). Δy = 3 – 3 = 0 Δx = 6 – (–1) = 7 More on Slopes Δy Δx = 7 4 m = Δy Δx = 0 7 Horizontal 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) = 7 Δx = 2 – (–2) = 4 m = Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 Two points are (–1, 3), (6, 3). Δy = 3 – 3 = 0 Δx = 6 – (–1) = 7 More on Slopes Δy Δx = 7 4 m = Δy Δx = 0 7 m = Δy Δx = 7 0 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) = 7 Δx = 2 – (–2) = 4 m = Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 Two points are (–1, 3), (6, 3). Δy = 3 – 3 = 0 Δx = 6 – (–1) = 7 More on Slopes Δy Δx = 7 4 m = Δy Δx = 0 7 m = Δy Δx = 7 0 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. More on Slopes
Lines that go through the quadrants I and III have positive slopes. More on 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. More on 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. More on 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. More on 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. More on 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 More on Slopes
Given a linear equation in x and y, solve for the variable y if possible, we get y = mx + b the number m is the slope and b is the y-intercept. More on Slopes
Given a linear equation in x and y, solve for the variable y if possible, we get y = mx + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only if the y-term is present. More on Slopes
Given a linear equation in x and y, solve for the variable y if possible, we get y = mx + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only if the y-term is present. More on 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 + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only if the y-term is present. More on 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 + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only if the y-term is present. More on 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 + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only if the y-term is present. More on 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 + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only 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 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 + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only 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). 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 + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only if the y-term is present. More on 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 + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only 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.
Given a linear equation in x and y, solve for the variable y if possible, we get y = mx + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only 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 + 6 More on Slopes
b. 0 = –2y + 6 solve for y More on Slopes
b. 0 = –2y + 6 solve for y 2y = 6 y = 3 More on Slopes
b. 0 = –2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3 More on Slopes
b. 0 = –2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3 Hence the slope m is 0. More on 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). More on 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. More on 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. More on 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 More on 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 More on 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 More on Slopes The variable y can’t be isolated because there is no y. Hence the slope is undefined and 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 More on Slopes The variable y can’t be isolated because there is no y. Hence the slope is undefined and 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 More on Slopes The variable y can’t be isolated because there is no y. Hence the slope is undefined and 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 More on Slopes The variable y can’t be isolated because there is no y. Hence the slope is undefined and this is a vertical line. Solve for x 3x = 6  x = 2. This is the vertical line x = 2.
Two Facts About Slopes I. Parallel lines have the same slope. More on Slopes
Two Facts About Slopes I. Parallel lines have the same slope. II. Slopes of perpendicular lines are the negative reciprocal of each other. More on Slopes
Two Facts About Slopes I. 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? More on Slopes
Two Facts About Slopes I. 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 More on Slopes
Two Facts About Slopes I. 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 More on Slopes
Two Facts About Slopes I. 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 More on Slopes
Two Facts About Slopes I. 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. More on Slopes
Two Facts About Slopes I. 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. More on Slopes
Two Facts About Slopes I. 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. More on Slopes b. What is the slope of L if L is perpendicular to 3x = 2y + 4?
Two Facts About Slopes I. 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. More on 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 Slopes I. 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. More on 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 = y2 3
Two Facts About Slopes I. 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. More on 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 Slopes I. 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. More on 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 Slopes I. 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. rise run=m = Δy Δx y2 – 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. More on 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) More on 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 = x F. 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. More on Slopes
36. The equation of the line is 3x = –5y+7 37. The equation of the line is 0 = –5y+7 38. The equation of the line is 3x = 7 39. The line is parallel to 2y = 5 – 6x 40. the line is perpendicular to 2y = 5 – 6x 41. 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 Slopes Find the slope, if possible of each of the following lines

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37 more on slopes-x

  • 1.
    Frank Ma ©2011 More on Slopes
  • 2.
  • 3.
    Definition of Slope Let(x1, y1) and (x2, y2) be two points on a line, (x1, y1) (x2, y2) More on Slopes
  • 4.
    Definition of Slope Let(x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δxm = (x1, y1) (x2, y2) More on Slopes
  • 5.
    Definition of Slope Let(x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx y2 – y1 x2 – x1 m = = (x1, y1) (x2, y2) More on Slopes
  • 6.
    Definition of Slope Let(x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx y2 – y1 x2 – x1 m = = Geometry of Slope (x1, y1) (x2, y2) More on Slopes
  • 7.
    Definition of Slope Let(x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx y2 – y1 x2 – x1 m = = (x1, y1) (x2, y2) Δy=y2–y1=rise Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points. More on Slopes
  • 8.
    Definition of Slope Let(x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx y2 – y1 x2 – x1 m = = (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. More on Slopes
  • 9.
    Definition of Slope Let(x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx y2 – y1 x2 – x1 m = = (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. Δy Δx =Therefore m is the ratio of the “rise” to the “run”. More on Slopes
  • 10.
    Definition of Slope Let(x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx y2 – y1 x2 – x1 m = = rise run= (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. Δy Δx =Therefore m is the ratio of the “rise” to the “run”. m = Δy Δx y2 – y1 x2 – x1 = More on Slopes
  • 11.
    Definition of Slope Let(x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx y2 – y1 x2 – x1 m = = rise run= (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. Δy Δx =Therefore m is the ratio of the “rise” to the “run”. m = Δy Δx y2 – y1 x2 – x1 = easy to memorize More on Slopes
  • 12.
    Definition of Slope Let(x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx y2 – y1 x2 – x1 m = = rise run= (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. Δy Δx =Therefore m is the ratio of the “rise” to the “run”. m = Δy Δx y2 – y1 x2 – x1 = easy to memorize the exact formula More on Slopes
  • 13.
    Definition of Slope Let(x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx y2 – y1 x2 – x1 m = = rise run= (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. Δy Δx =Therefore m is the ratio of the “rise” to the “run”. m = Δy Δx y2 – y1 x2 – x1 = easy to memorize the exact formula geometric meaning More on Slopes
  • 14.
    Example A. Findthe slope of each of the following lines. More on Slopes
  • 15.
    Example A. Findthe slope of each of the following lines. Two points are (–3, 1), (4, 1). More on Slopes
  • 16.
    Example A. Findthe slope of each of the following lines. Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 More on Slopes
  • 17.
    Example A. Findthe slope of each of the following lines. Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 More on Slopes
  • 18.
    Example A. Findthe slope of each of the following lines. Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 More on Slopes m = Δy Δx = 0 7 = 0
  • 19.
    Example A. Findthe slope of each of the following lines. Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 More on Slopes m = Δy Δx = 0 7 Horizontal line Slope = 0 = 0
  • 20.
    Example A. Findthe slope of each of the following lines. Two points are (–2, –4), (2, 3). Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 More on Slopes m = Δy Δx = 0 7 Horizontal line Slope = 0 = 0
  • 21.
    Example A. Findthe 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) = 0 Δx = 4 – (–3) = 7 More on Slopes m = Δy Δx = 0 7 Horizontal line Slope = 0 = 0
  • 22.
    Example A. Findthe slope of each of the following lines. Two points are (–2, –4), (2, 3). Δy = 3 – (–4) = 7 Δx = 2 – (–2) = 4 Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 More on Slopes m = Δy Δx = 0 7 Horizontal line Slope = 0 = 0
  • 23.
    Example A. Findthe slope of each of the following lines. Two points are (–2, –4), (2, 3). Δy = 3 – (–4) = 7 Δx = 2 – (–2) = 4 m = Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 More on Slopes Δy Δx = 7 4 m = Δy Δx = 0 7 Horizontal line Slope = 0 = 0
  • 24.
    Example A. Findthe slope of each of the following lines. Two points are (–2, –4), (2, 3). Δy = 3 – (–4) = 7 Δx = 2 – (–2) = 4 m = Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 More on Slopes Δy Δx = 7 4 m = Δy Δx = 0 7 Horizontal line Slope = 0 Tilted line Slope = 0 = 0
  • 25.
    Example A. Findthe slope of each of the following lines. Two points are (–2, –4), (2, 3). Δy = 3 – (–4) = 7 Δx = 2 – (–2) = 4 m = Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 Two points are (–1, 3), (6, 3). More on Slopes Δy Δx = 7 4 m = Δy Δx = 0 7 Horizontal line Slope = 0 Tilted line Slope = 0 = 0
  • 26.
    Example A. Findthe slope of each of the following lines. Two points are (–2, –4), (2, 3). Δy = 3 – (–4) = 7 Δx = 2 – (–2) = 4 m = Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 Two points are (–1, 3), (6, 3). Δy = 3 – 3 = 0 More on Slopes Δy Δx = 7 4 m = Δy Δx = 0 7 Horizontal line Slope = 0 Tilted line Slope = 0 = 0
  • 27.
    Example A. Findthe slope of each of the following lines. Two points are (–2, –4), (2, 3). Δy = 3 – (–4) = 7 Δx = 2 – (–2) = 4 m = Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 Two points are (–1, 3), (6, 3). Δy = 3 – 3 = 0 Δx = 6 – (–1) = 7 More on Slopes Δy Δx = 7 4 m = Δy Δx = 0 7 Horizontal line Slope = 0 Tilted line Slope = 0 = 0
  • 28.
    Example A. Findthe slope of each of the following lines. Two points are (–2, –4), (2, 3). Δy = 3 – (–4) = 7 Δx = 2 – (–2) = 4 m = Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 Two points are (–1, 3), (6, 3). Δy = 3 – 3 = 0 Δx = 6 – (–1) = 7 More on Slopes Δy Δx = 7 4 m = Δy Δx = 0 7 m = Δy Δx = 7 0 Horizontal line Slope = 0 Tilted line Slope = 0 = 0 (UDF)
  • 29.
    Example A. Findthe slope of each of the following lines. Two points are (–2, –4), (2, 3). Δy = 3 – (–4) = 7 Δx = 2 – (–2) = 4 m = Two points are (–3, 1), (4, 1). Δy = 1 – (1) = 0 Δx = 4 – (–3) = 7 Two points are (–1, 3), (6, 3). Δy = 3 – 3 = 0 Δx = 6 – (–1) = 7 More on Slopes Δy Δx = 7 4 m = Δy Δx = 0 7 m = Δy Δx = 7 0 Horizontal line Slope = 0 Vertical line Slope is UDF Tilted line Slope = 0 = 0 (UDF)
  • 30.
    Lines that gothrough the quadrants I and III have positive slopes. More on Slopes
  • 31.
    Lines that gothrough the quadrants I and III have positive slopes. More on Slopes III III IV
  • 32.
    Lines that gothrough the quadrants I and III have positive slopes. Lines that go through the quadrants II and IV have negative slopes. More on Slopes III III IV
  • 33.
    Lines that gothrough the quadrants I and III have positive slopes. Lines that go through the quadrants II and IV have negative slopes. More on Slopes III III IV III III IV
  • 34.
    Lines that gothrough the quadrants I and III have positive slopes. Lines that go through the quadrants II and IV have negative slopes. More on Slopes The formula for slopes requires geometric information, i.e. the positions of two points on the line. III III IV III III IV
  • 35.
    Lines that gothrough the quadrants I and III have positive slopes. Lines that go through the quadrants II and IV have negative slopes. More on 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
  • 36.
    Given a linearequation in x and y, solve for the variable y if possible, we get y = mx + b More on Slopes
  • 37.
    Given a linearequation in x and y, solve for the variable y if possible, we get y = mx + b the number m is the slope and b is the y-intercept. More on Slopes
  • 38.
    Given a linearequation in x and y, solve for the variable y if possible, we get y = mx + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only if the y-term is present. More on Slopes
  • 39.
    Given a linearequation in x and y, solve for the variable y if possible, we get y = mx + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only if the y-term is present. More on 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.
  • 40.
    Given a linearequation in x and y, solve for the variable y if possible, we get y = mx + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only if the y-term is present. More on 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.
  • 41.
    Given a linearequation in x and y, solve for the variable y if possible, we get y = mx + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only if the y-term is present. More on 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.
  • 42.
    Given a linearequation in x and y, solve for the variable y if possible, we get y = mx + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only if the y-term is present. More on 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.
  • 43.
    Given a linearequation in x and y, solve for the variable y if possible, we get y = mx + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only 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 Example B. Write the equations into the slope intercept form, list the slopes, the y-intercepts and draw the lines.
  • 44.
    Given a linearequation in x and y, solve for the variable y if possible, we get y = mx + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only 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). Example B. Write the equations into the slope intercept form, list the slopes, the y-intercepts and draw the lines.
  • 45.
    Given a linearequation in x and y, solve for the variable y if possible, we get y = mx + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only if the y-term is present. More on 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).
  • 46.
    Given a linearequation in x and y, solve for the variable y if possible, we get y = mx + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only 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.
  • 47.
    Given a linearequation in x and y, solve for the variable y if possible, we get y = mx + b the number m is the slope and b is the y-intercept. This is called the slope intercept form and this can be done only 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.
  • 48.
    b. 0 =–2y + 6 More on Slopes
  • 49.
    b. 0 =–2y + 6 solve for y More on Slopes
  • 50.
    b. 0 =–2y + 6 solve for y 2y = 6 y = 3 More on Slopes
  • 51.
    b. 0 =–2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3 More on Slopes
  • 52.
    b. 0 =–2y + 6 solve for y 2y = 6 y = 3 y = 0x + 3 Hence the slope m is 0. More on Slopes
  • 53.
    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). More on Slopes
  • 54.
    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. More on Slopes
  • 55.
    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. More on Slopes
  • 56.
    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 More on Slopes
  • 57.
    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 More on Slopes The variable y can’t be isolated because there is no y.
  • 58.
    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 More on Slopes The variable y can’t be isolated because there is no y. Hence the slope is undefined and this is a vertical line.
  • 59.
    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 More on Slopes The variable y can’t be isolated because there is no y. Hence the slope is undefined and this is a vertical line. Solve for x 3x = 6  x = 2.
  • 60.
    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 More on Slopes The variable y can’t be isolated because there is no y. Hence the slope is undefined and this is a vertical line. Solve for x 3x = 6  x = 2. This is the vertical line x = 2.
  • 61.
    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 More on Slopes The variable y can’t be isolated because there is no y. Hence the slope is undefined and this is a vertical line. Solve for x 3x = 6  x = 2. This is the vertical line x = 2.
  • 62.
    Two Facts AboutSlopes I. Parallel lines have the same slope. More on Slopes
  • 63.
    Two Facts AboutSlopes I. Parallel lines have the same slope. II. Slopes of perpendicular lines are the negative reciprocal of each other. More on Slopes
  • 64.
    Two Facts AboutSlopes I. 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? More on Slopes
  • 65.
    Two Facts AboutSlopes I. 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 More on Slopes
  • 66.
    Two Facts AboutSlopes I. 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 More on Slopes
  • 67.
    Two Facts AboutSlopes I. 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 More on Slopes
  • 68.
    Two Facts AboutSlopes I. 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. More on Slopes
  • 69.
    Two Facts AboutSlopes I. 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. More on Slopes
  • 70.
    Two Facts AboutSlopes I. 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. More on Slopes b. What is the slope of L if L is perpendicular to 3x = 2y + 4?
  • 71.
    Two Facts AboutSlopes I. 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. More on 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
  • 72.
    Two Facts AboutSlopes I. 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. More on 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 = y2 3
  • 73.
    Two Facts AboutSlopes I. 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. More on 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
  • 74.
    Two Facts AboutSlopes I. 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. More on 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
  • 75.
    Summary on Slopes Howto Find Slopes I. 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. rise run=m = Δy Δx y2 – y1 x2 – x1 =
  • 76.
    Exercise A. Identifythe 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. More on Slopes
  • 77.
    16. 17. 18. ExerciseC. 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) More on Slopes
  • 78.
    Exercise D. 34. Identifywhich 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 = x F. 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. More on Slopes
  • 79.
    36. The equationof the line is 3x = –5y+7 37. The equation of the line is 0 = –5y+7 38. The equation of the line is 3x = 7 39. The line is parallel to 2y = 5 – 6x 40. the line is perpendicular to 2y = 5 – 6x 41. 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 Slopes Find the slope, if possible of each of the following lines