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

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**Write an equation given the slope and y-intercept**

EXAMPLE 1 Write an equation given the slope and y-intercept SOLUTION 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 3 4 y = x (–2) 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 3. m = – , b = 3 4 7 2 ANSWER ANSWER y = – x + 3 4 7 2 y = x + 1 3 2. m = –2 , b = –4 ANSWER y = –2x – 4

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**Write an equation given the slope and a point**

EXAMPLE 2 Write an equation given the slope and a point Write an equation of the line that passes through (5, 4) and has a gradient of –3. SOLUTION Because you know the slope and a point on the line, use point-slope form to write an equation of the line. Let (x1, y1) = (5, 4) and m = –3. y – y1 = m(x – x1) Use point-slope form. y – 4 = –3(x – 5) Substitute for m, x1, and y1. y – 4 = –3x + 15 Distributive property y = –3x + 19 Write in slope-intercept form.

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EXAMPLE 3 Write equations of parallel or perpendicular lines 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 m1 = –4. So, a line parallel to it has a slope of m2 = m1 = –4. You know the slope and a point on the line, so use the point-slope form with (x1, y1) = (–2, 3) to write an equation of the line.

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**Write equations of parallel or perpendicular lines**

EXAMPLE 3 Write equations of parallel or perpendicular lines y – y1 = m2(x – x1) Use point-slope form. y – 3 = –4(x – (–2)) Substitute for m2, x1, and y1. y – 3 = –4(x + 2) Simplify. y – 3 = –4x – 8 Distributive property y = –4x – 5 Write in slope-intercept form.

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**Write equations of parallel or perpendicular lines**

EXAMPLE 3 Write equations of parallel or perpendicular lines b. A line perpendicular to a line with slope m1 = –4 has a slope of m2 = – = . Use point-slope form with (x1, y1) = (–2, 3) 1 4 m1 y – y1 = m2(x – x1) Use point-slope form. y – 3 = (x – (–2)) 1 4 Substitute for m2, x1, and y1. y – 3 = (x +2) 1 4 Simplify. y – 3 = x + 1 4 2 Distributive property Write in slope-intercept form.

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GUIDED PRACTICE GUIDED PRACTICE for Examples 2 and 3 4. Write an equation of the line that passes through (–1, 6) and has a slope of 4. ANSWER y = 4x + 10 5. 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. ANSWER y = 3x – 14

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EXAMPLE 4 Write an equation given two points Write an equation of the line that passes through (5, –2) and (2, 10). SOLUTION The line passes through (x1, y1) = (5,–2) and (x2, y2) = (2, 10). Find its slope. y2 – y1 m = x2 – x1 10 – (–2) = 2 – 5 12 –3 = –4

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**Write an equation given two points**

EXAMPLE 4 Write an equation given two points 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 (x1, y1) = (4, – 7). y2 – y1 = m(x – x1) Use point-slope form. y – 10 = – 4(x – 2) Substitute for m, x1, and y1. y – 10 = – 4x + 8 Distributive property y = – 4x + 8 Write in slope-intercept form.

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