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Writing Equations of a Line

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Various Forms of an Equation of a Line. Gradient-Intercept Form Standard Form Point-Gradient Form

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Write an equation given the slope and y-intercept EXAMPLE 1 Write an equation of the line shown.

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SOLUTION Write an equation given the slope and y-intercept EXAMPLE 1 From the graph, you can see that the slope is m = and the y- intercept is c = –2. Use slope-intercept form to write an equation of the line. 3 4 y = mx + c Use gradient-intercept form. y = x + (–2) 3 4 Substitute for m and –2 for b. 3 4 y = x (–2) 3 4 Simplify.

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GUIDED PRACTICE for Example 1 Write an equation of the line that has the given slope and y- intercept. 1. m = 3, b = 1 y = x ANSWER 2. m = –2, b = –4 y = –2x – 4 ANSWER 3. m = –, b = y = – x ANSWER

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Write an equation given the slope and a point EXAMPLE 2 Write an equation of the line that passes through (5, 4) and has a gradient of –3. Because you know the slope and a point on the line, use point-slope form to write an equation of the line. Let (x 1, y 1 ) = (5, 4) and m = –3. y – y 1 = m(x – x 1 ) Use point-slope form. y – 4 = –3(x – 5) Substitute for m, x 1, and y 1. y – 4 = –3x + 15 Distributive property SOLUTION y = –3x + 19 Write in slope-intercept form.

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EXAMPLE 3 Write an equation of the line that passes through (–2,3) and is (a) parallel to, and (b) perpendicular to, the line y = –4x + 1. SOLUTION a. The given line has a slope of m 1 = –4. So, a line parallel to it has a slope of m 2 = m 1 = –4. You know the slope and a point on the line, so use the point- slope form with (x 1, y 1 ) = (–2, 3) to write an equation of the line. Write equations of parallel or perpendicular lines

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EXAMPLE 3 y – 3 = –4(x – (–2)) y – y 1 = m 2 (x – x 1 ) Use point-slope form. Substitute for m 2, x 1, and y 1. y – 3 = –4(x + 2) Simplify. y – 3 = –4x – 8 Distributive property y = –4x – 5 Write in slope-intercept form. Write equations of parallel or perpendicular lines

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EXAMPLE 3 b. A line perpendicular to a line with slope m 1 = –4 has a slope of m 2 = – =. Use point-slope form with (x 1, y 1 ) = (–2, 3) m1m1 y – y 1 = m 2 (x – x 1 ) Use point-slope form. y – 3 = (x – (–2)) 1 4 Substitute for m 2, x 1, and y 1. y – 3 = (x +2) 1 4 Simplify. y – 3 = x Distributive property Write in slope-intercept form. Write equations of parallel or perpendicular lines

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GUIDED PRACTICE for Examples 2 and 3 GUIDED PRACTICE 4. Write an equation of the line that passes through (–1, 6) and has a slope of 4. y = 4x Write an equation of the line that passes through (4, –2) and is (a) parallel to, and (b) perpendicular to, the line y = 3x – 1. y = 3x – 14 ANSWER

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Write an equation given two points EXAMPLE 4 Write an equation of the line that passes through (5, –2) and (2, 10). SOLUTION The line passes through (x 1, y 1 ) = (5,–2) and (x 2, y 2 ) = (2, 10). Find its slope. y 2 – y 1 m =m = x 2 – x 1 10 – (–2) = 2 – 5 12 –3 == –4

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Write an equation given two points EXAMPLE 4 You know the slope and a point on the line, so use point-slope form with either given point to write an equation of the line. Choose (x 1, y 1 ) = (4, – 7). y 2 – y 1 = m(x – x 1 ) Use point-slope form. y – 10 = – 4(x – 2) Substitute for m, x 1, and y 1. y – 10 = – 4x + 8 Distributive property Write in slope-intercept form. y = – 4x + 8

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