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§ 2.4 The Slope of a Line

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Slope Slope of a Line

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**Slope Example: Find the slope of the line through (4, -3) and (2, 2)**

If we let (x1, y1) be (4, -3) and (x2, y2) be (2, 2), then Note: If we let (x1, y1) be (2, 2) and (x2, y2) be (4, -3), then we get the same result.

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**Slope-Intercept Form Slope-Intercept Form of a line**

y = mx + b has a slope of m and has a y-intercept of (0, b). This form is useful for graphing, since you have a point and the slope readily visible.

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**Slope-Intercept Form Example:**

Find the slope and y-intercept of the line –3x + y = -5. First, we need to solve the linear equation for y. By adding 3x to both sides, y = 3x – 5. Once we have the equation in the form of y = mx + b, we can read the slope and y-intercept. slope is 3 y-intercept is (0, – 5)

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**Slope-Intercept Form Example:**

Find the slope and y-intercept of the line 2x – 6y = 12. First, we need to solve the linear equation for y. – 6y = – 2x Subtract 2x from both sides. y = x – Divide both sides by – 6. Since the equation is now in the form of y = mx + b, slope is y-intercept is (0, –2)

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**Slope of a Horizontal Line**

For any 2 points, the y values will be equal to the same real number. The numerator in the slope formula = 0 (the difference of the y-coordinates), but the denominator 0 (two different points would have two different x-coordinates). So the slope = 0.

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**Slope of a Vertical Line**

For any 2 points, the x values will be equal to the same real number. The denominator in the slope formula = 0 (the difference of the x-coordinates), but the numerator 0 (two different points would have two different y-coordinates), So the slope is undefined (since you can’t divide by 0).

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**Summary of Slope of Lines**

If a line moves up as it moves from left to right, the slope is positive. If a line moves down as it moves from left to right, the slope is negative. Horizontal lines have a slope of 0. Vertical lines have undefined slope (or no slope).

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Parallel Lines Two lines that never intersect are called parallel lines. Parallel lines have the same slope unless they are vertical lines, which have no slope. Vertical lines are also parallel.

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**Parallel Lines Example:**

Find the slope of a line parallel to the line passing through (0,3) and (6,0) So the slope of any parallel line is also –½

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Perpendicular Lines Two lines that intersect at right angles are called perpendicular lines Two nonvertical perpendicular lines have slopes that are negative reciprocals of each other The product of their slopes will be –1 Horizontal and vertical lines are perpendicular to each other

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**Perpendicular Lines Example:**

Find the slope of a line perpendicular to the line passing through (-1,3) and (2,-8) So the slope of any perpendicular line is

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**Parallel and Perpendicular Lines**

Example: Determine whether the following lines are parallel, perpendicular, or neither. –5x + y = –6 and x + 5y = 5 First, we need to solve both equations for y. In the first equation, y = 5x – Add 5x to both sides. In the second equation, 5y = –x Subtract x from both sides. y = x Divide both sides by 5. The first equation has a slope of 5 and the second equation has a slope of , so the lines are perpendicular.

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