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Writing Equations of a Line

Discover the minimum information required for writing equations of lines and explore the Slope-Intercept Form, Standard Form, and Point-Slope Form. Learn to write equations using slope and y-intercept or slope and a point. Practice writing equations for parallel or perpendicular lines.

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Writing Equations of a Line

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  1. Writing Equations of a Line What is the minimum information needed?

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

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

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

  5. 7 7 3 3. m = – , b = 2 2 4 3 y =– x + 4 y =x + 1 3 for Example 1 GUIDED PRACTICE Write an equation of the line that has the given slope and y-intercept. 1. m = 3, b = 1 ANSWER ANSWER 2. m = –2 , b = –4 ANSWER y = –2x – 4

  6. 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 slope 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.

  7. 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.

  8. 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, andy1. y – 3 =–4(x + 2) Simplify. y – 3 =–4x – 8 Distributive property y =–4x – 5 Write in slope-intercept form.

  9. 1 1 1 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) 4 2 4 1 m1 y –3= (x – (–2)) 1 y – 3 = (x +2) 4 1 4 y – 3 = x + EXAMPLE 3 Write equations of parallel or perpendicular lines y –y1=m2(x –x1) Use point-slope form. Substitute for m2, x1, andy1. Simplify. Distributive property Write in slope-intercept form.

  10. for Examples 2 and 3 GUIDED PRACTICE GUIDED PRACTICE 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. y = 3x – 14 ANSWER

  11. 12 –3 10 – (–2) y2 – y1 = m= = = –4 x2 – x1 2 –5 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.

  12. 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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