Solve by the addition method

Question

Solve by the addition method: 3x - 2y = -7

                                            4x + 3y = 2

Answer 1

The Elimination Method (also called the Addition Method) adds/subtracts two equation in order to eliminate one of the variables so that we can solve for the other variable.  In order for the coefficients to add/subtract to zero, they must be the same (or opposite); this means that we might have to first multiply one or both equations in order to find a common multiple for that coefficient.

 

Here is this problem:

   3x  - 2y = -7                 [eq1]

   4x + 3y =  2                 [eq2]

 

We make either the x-coefficient or the y-coefficient the same or opposite.  Let's make the y-coefficient opposites and then add the equations:

    9x  -  6y  =  -21        [note: multiply all terms in eq1 by 3]

    8x  + 6y  =     4        [note: multiply all terms in eq2 by 2]

  ---------------------     [elimination; add equations]

    17x         =  -17

      x           =  -1       [call this eq3]

 

Now, we may either substitute (-1) for x in either equation to solve for y; or we may use the elimination method again:

   4x + 3y =  2          [eq2 again;  you may select either eq1 or eq2]

 - 4x         =  4        [note:  multiply all terms in eq3 by (-4)]

 ------------------      [elimination method; add equations]

          3y   =  6

           y    =  2          [divide both sides by 3]

 

Checking (very important):

  Is  3(-1) - 2(2) = -7   ?

        -3    -4       = -7   ?

            -7  =  -7   ?yes

  Is 4(-1) + 3(2) = 2   ?

       -4  + 6  = 2   ?

             2  =  2   ?yes