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**Equations of Lines and Linear Models**

Section 2.4 Equations of Lines and Linear Models

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**Objectives Point-Slope Form Horizontal and Vertical Lines**

Parallel and Perpendicular Lines

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POINT-SLOPE FORM The line with slope m passing through the point (x1, y1) is given by y = m(x – x1) + y1 or equivalently, y – y1 = m(x – x1), the point-slope form of a line.

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Example Find the point-slope form of a line passing through the point (3, 1) with slope 2. Does the point (4, 3) lie on this line? Solution Let m = 2 and (x1, y1) = (3,1) in the point-slope form. To determine whether the point (4, 3) lies on the line, substitute 4 for x and 3 for y. y – y1 = m(x – x1) y − 1 = 2(x – 3) 3 – 1 ? 2(4 – 3) 2 = 2 The point (4, 3) lies on the line because it satisfies the point-slope form.

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Example Use the point-slope form to find an equation of the line passing through the points (−2, 3) and (2, 5). Solution Before we can apply the point-slope form, we must find the slope.

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Example (cont) We can use either (−2, 3) or (2, 5) for (x1, y1) in the point-slope form. If we choose (−2, 3), the point-slope form becomes the following. If we choose (2, 5), the point-slope form with x1 = 2 and y1 = 5 becomes y – y1= m(x – x1)

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Example Find an equation of the line passing through (4, 2) and (1, 5). Solution First find the slope of the line.

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Example (cont) Find an equation of the line passing through (4, 2) and (1, 5). Solution Now substitute −1 for m and (4, 2) for x and y in the slope intercept form. The point (1, 5) could be used instead. The slope-intercept form is y = −x + 6.

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Example Find a line y = mx + b that models the data in the table. Solution Carefully make a scatter plot of the data and sketch a line. x 1 2 3 4 5 f(x) 6 11 18 24 31

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Example (cont) Find a line y = mx + b that models the data in the table. Solution Chose two points on the line and find the equation. The equation of the line is y = 6(x – 3) + 18 or y = 6x. x 1 2 3 4 5 f(x) 6 11 18 24 31

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**The equation of a horizontal line with y-intercept b is y = b.**

EQUATIONS OF HORIZONTAL AND VERTICAL LINES The equation of a horizontal line with y-intercept b is y = b. The equation of a vertical line with x-intercept h is x = h.

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Example Find equations of the vertical and horizontal lines that pass through the point (−5, 2). Graph these two lines. Solution The x-coordinate of the point (−5, 2) is −5. The vertical line x = −5 passes through every point on the xy-plane including the point (−5, 2). Similarly, the horizontal line y = 2 passes through every point with a y-coordinate of 2 including the point (−5, 2).

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Example (cont) Find equations of the vertical and horizontal lines that pass through the point (−5, 2). Graph these two lines. Solution x = −5 y = 2

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**PARALLEL LINES Two lines with the same slope are parallel.**

Two nonvertical parallel lines have the same slope.

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Example Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2, 1). Solution The line has a slope of 3 any parallel line also has slope 3.

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PERPENDICULAR LINES Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = −1. If two lines have slopes m1 and m2 such that m1 · m2 = −1, then they are perpendicular. A vertical and horizontal line are perpendicular.

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Example Find the slope-intercept form of the line perpendicular to y = x – 3 passing through the point (4, 6). Solution The line has slope m1 = 1. The slope of the perpendicular line is m2 = −1. The slope-intercept form of a line having slope −1 and passing through (4, 6) can be found as follows.

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Example Find the slope-intercept form of each line shown. Verify that the two lines are perpendicular. Solution The graph of y1 has the slope m1 = 3/2 because the line rises 3 units for every 2 units of run. It’s y-intercept is −1/2, and it’s slope intercept form is The graph of y2 has the slope m2 = −2/3 because the line falls 2 units for every 3 units of run. It’s y-intercept is 1/3, and it’s slope intercept form is y2 y1

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Example (cont) Find the slope-intercept form of each line shown. Verify that the two lines are perpendicular. Solution The graph of y2 has the slope m2 = −2/3 because the line falls 2 units for every 3 units of run. It’s y-intercept is 1/3, and it’s slope intercept form is y2 y1

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