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3.3 The Slope of a Line

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An important characteristic of the lines we graphed in Section 3.2 is their slant, or “steepness.” One way to measure the steepness of a line is to compare the vertical change in the line with the horizontal change while moving along the line from one fixed point to another. This measure of steepness is called the slope of the line. The Slope of a Line Slide 3.3-3

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Objective 1 Find the slope of a line, given two points. Slide 3.3-4

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To find the steepness, or slope, of the line in the figure below, begin at point Q and move to point P. The vertical change, or rise, is the change in the y- values, which is the difference 6 − 1 = 5 units. The horizontal change, or run, is the change in the x-values, which is the difference 5 − 2 = 3 units. The slope is the ratio of the vertical change in y to the horizontal change in x. Count squares on the grid to find the change. Upward and rightward movements are positive. Downward and leftward movements are negative. Slide Find the slope of a line, given two points.

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Solution: Find the slope of the line. Slide Finding the Slope of a Line CLASSROOM EXAMPLE 1

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Slope Formula The slope m of a line through the points (x 1, y 1 ) and (x 2, y 2 ) is The slope of a line can be found through two nonspecific points. This notation is called subscript notation, read x 1 as “x-sub-one” and x 2 as “x-sub- two”. Moving along the line from the point (x 1, y 1 ) to the point (x 2, y 2 ), we see that y changes by y 2 − y 1 units. This is the vertical change (rise). Similarly, x changes by x 2 − x 1 units, which is the horizontal change (run). The slope of the line is the ratio of y 2 − y 1 to x 2 − x 1. Slide Find the slope of a line, given two points. (cont’d) The slope of a line is the same for any two points on the line.

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Solution: Find the slope of the line through (6, − 8) and (−2, 4). and yield the same slope. Make sure to start with the x- and y-values of the same point and subtract the x- and y-values of the other point. Slide Finding Slopes of Lines CLASSROOM EXAMPLE 2

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Orientation of Lines with Positive and Negative Slopes A line with a positive slope rises (slants up) from left to right. A line with a negative slope falls (slants down) from left to right. Slopes of Horizontal and Vertical Lines Horizontal lines, with equations of the form y = k, have slope 0. Vertical lines, with equations of the form x = k, have undefined slopes. Slide Find the slope of a line, given two points. (cont’d)

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Solution: Find the slope of the line through (2, 5) and (−1, 5). Slide Finding the Slope of a Horizontal Line CLASSROOM EXAMPLE 3

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Solution: Find the slope of the line through (3, 1) and (3,−4). undefined slope Slide Finding the Slope of a Vertical Line CLASSROOM EXAMPLE 4

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Objective 2 Find the slope from the equation of a line. Slide

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Consider the equation y = −3x + 5. The slope of the line can be found by choosing two different points for value x and then solving for the corresponding values of y. We choose x = −2 and x = 4. The ordered pairs are (−2,11) and (4, −7). Now we use the slope formula. Slide Find the slope from the equation of a line.

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Step 1: Solve the equation for y. Step 2: The slope is given by the coefficient of x. The slope, −3 is found, which is the same number as the coefficient of x in the given equation y = −3x + 5. It can be shown that this always happens, as long as the equation is solved for y. Finding the Slope of a Line from Its Equation Slide Find the slope from the equation of a line. (cont’d)

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Solution: Find the slope of the line 3x + 2y = 9. Slide Finding Slopes from Equations CLASSROOM EXAMPLE 5

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Objective 3 Use slopes to determine whether two lines are parallel, perpendicular, or neither. Slide

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Two lines in a plane that never intersect are parallel. We use slopes to tell whether two lines are parallel. Nonvertical parallel lines always have equal slopes. Use slopes to determine whether two lines are parallel, perpendicular, or neither. Lines are perpendicular if they intersect at a 90° angle. The product of the slopes of two perpendicular lines, neither of which is vertical, is always − 1. This means that the slopes of perpendicular lines are negative (or opposite) reciprocals — if one slope is the nonzero number a, the other is. The table to the right shows several examples. Slide

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Slopes of Parallel and Perpendicular Lines Two lines with the same slope are parallel. Two lines whose slopes have a product of − 1 are perpendicular. Slide Use slopes to determine whether two lines are parallel, perpendicular, or neither. (cont’d)

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Solution: The product of their slopes is − 1, so they are perpendicular Decide whether the pair of lines is parallel, perpendicular, or neither. Slide Deciding Whether Two Lines Are Parallel or Perpendicular CLASSROOM EXAMPLE 6

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