Curve Intersection Intersection of Lines:  Substitution vs Elimination Recall that if we want to find the intersection point of two lines, we have two choices:  substitution and elimination. Example:  (Substitution) Solve         x + 2y = 5         4x - 3y = -2 We use the first equation to solve for x:         x = 5 - 2y then we plug  this into the second equation to get         4(5 - 2y) - 3y = -2         -11y + 20 = -2         y = 2 stick this back into the equation for x to get:         x = 5 - 2(2) = 1. Example: (Elimination) Solve           2x + 5y = 19         3x - 5y = -9 We add the equation to get         5x = 10,          x = 2 Hence          2(2) + 5y = 19         y = 3. Intersection of Other Curves Example:  Substitution Find the intersection of the curves         x2 + y2 = 25  and          y = 1/3 x + 3 We use the method of substitution to arrive at         x2 + (1/3 x + 3)2 = 25         x2 + 1/9 x2 + 2x + 9 = 25         10x2 + 18x - 144 = 0         5x2 + 9x - 72 = 0         (5x + 24)(x - 3) = 0         x = -24/5     or     x = 3         y = (1/3)(-24/5 )+ 3 or y = 1/3(3) + 3         y = -7/5 or 4. We get the points          (-24/5,-7/5)     and     (3,4) Example:  Elimination         x2 + 2y2 = 18         2x2 + y2 = 15 We multiply the first equation by 2 and subtract the second equation to get:         3y2 = 21         y2 = 7         y =      or     y = - Substituting back into the first equation, we get:         x2 + 2(7) = 18         x = 2     or     x = -2,  hence we get the four points:         (2,),     (-2,),     (2,-),     (-2,-). Example:  Using a Graphing Calculator We will use a graphing calculator to find the intersection of         y2 + 16x = 0  and          y2 + 9x2 -18x = 18. To find the intersection we just use the intersection function on the graphing calculator.          Example We will use the intercept method to solve         (x - 7)(x + 4) = (x + 1)2   We find the intersection of the two curves          y = (x - 7)(x + 4) and           y = (x + 1)2     Back to Math 103A Home Page Back to the Math Department Home e-mail Questions and Suggestions