Curve Intersection

  1. 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.


  2. 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  

     

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