Topic 4.2 (The Equation of a Line)

      To determine the equation of a line the following formulas are used.         

                     m  =  (y₂ - y₁) /  (x₂ - x₁)                     Slope formula

                     y = mx + b                                            Slope, y-intercept formula

                     y - y₁ = m(x - x₁)                                  Point Slope formula

                     Ax + By = C                                          Standard Form

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Sample problems

    Find the equation (point-slope form) of the line that passes through the

                      point  (4, 1)  and has slope  2.                         y - y₁ = m(x - x₁)

                                                 y  -  1  =  2(x  -  4)                       Ans.

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    Find the equation (point-slope form) of the line that passes through the

                        points (2, 3) & (4, 7).

               First find the slope:   m  =  (7 - 3) / (4 - 2)    =    4/2     =    2/1

             Use the point slope formula with the slope and one of the points.

                     (2, 3),       m = 2            y - 3 = 2(x - 2)                 Ans.

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    Find the equation (standard form) of the line that passes through the points

                      (3, 1) & (5, -2).

          First find the slope m  =   (-2 - 1) / (5 - 3) =    -3/2     Use the point slope, with the

                                                                                                       slope and one point.

             (3, 1),       m = -3/2        (y - 1) = (-3/2)(x - 3)            Multiply by 2

                                                    2(y - 1) = (-3)(x - 3)

                                                    2y - 2 = -3x + 9                 Put in standard form and there

         Ax + By = C              3x + 2y = 11      Ans.               should be no fraction in the

                                                                                                answer and A must be positive.

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          Find the equation (slope-intercept form) of the line that passes through the

                    point   (-4, 6)   and has slope   3.

          Use the point-slope formula:       y - 6 = 3[x - (-4)]               Solve for y.

                                                                      y - 6 = 3(x + 4)

                                                                      y - 6 = 3x + 12

          y = mx + b                                           y = 3x + 18          Ans

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          Find the equation of the horizontal and vertical line that passes through

                             the point (1, -3)

                                         x =  1                      Vertical line

                                         y = -3                      Horizontal Line

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          Find the equations of the line that passes through the point (-8, 11) and

                             has slope   -7/4.

          Use the point slope formula:       y - 11 = (-7/4)(x + 8)       multiply by 4

                                                                     4y - 44 = -7x - 56

                                                                             4y = -7x - 12

                                                            y - 11 = (-7/4)(x + 8)       point-slope form

                                                                  y = (-7/4) x - 3            slope-intercept form

                                                                    7x + 4y = -12            Standard Form