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Writing Equations of a Line
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Writing Equations of a Line

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  • 1. Writing Equations of a Line
  • 2. Various Forms of an Equation of a Line. Slope-Intercept Form Standard Form Point-Slope Form slope of the line intercept y mx b m b y = + = = − , , and are integers 0, must be postive Ax By C A B C A A + = > ( ) ( ) 1 1 1 1 slope of the line , is any point y y m x x m x y − = + = - -
  • 3. KEY CONCEPT Writing an Equation of a Line – Given slope m and y-intercept b • Use slope-intercept form y=mx+b – Given slope m and a point (x1,y1) • Use point-slope form – y - y1 = m ( x – x1) • Given points (x1,y1) and (x2,y2) – Find your slope then use point-slope form with either point.
  • 4. Write an equation given the slope and y-interceptEXAMPLE 1 Write an equation of the line shown.
  • 5. SOLUTION Write an equation given the slope and y-interceptEXAMPLE 1 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 4 y = mx + b Use slope-intercept form. y = x + (–2) 3 4 Substitute for m and –2 for b. 3 4 y = x –2 3 4 Simplify.
  • 6. 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 + 13 ANSWER 2. m = –2 , b = –4 y = –2x – 4 ANSWER 3. m = – , b =3 4 7 2 y = – x +3 4 7 2 ANSWER
  • 7. Write an equation given the slope and a pointEXAMPLE 2 Write an equation of the line that passes through (5, 4) and has a slope 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 (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 SOLUTION y = –3x + 19 Write in slope-intercept form.
  • 8. 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 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. Write equations of parallel or perpendicular lines
  • 9. EXAMPLE 3 y – 3 = –4(x – (–2)) y – y1 = m2(x – x1) Use point-slope form. 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. Write equations of parallel or perpendicular lines
  • 10. EXAMPLE 3 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 1 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 1 2 Distributive property Write in slope-intercept form. Write equations of parallel or perpendicular lines 1 7 4 2 y x= +
  • 11. GUIDED PRACTICE for Examples 2 and 3GUIDED PRACTICE 4. Write an equation of the line that passes through (–1, 6) and has a slope of 4. 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 – 14ANSWER ANSWER
  • 12. Write an equation given two pointsEXAMPLE 4 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
  • 13. Write an equation given two pointsEXAMPLE 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 (x1, y1) = (2, 10). 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 Write in slope-intercept form.y = – 4x + 8