Solve the system using the elimination method

Question

-2x + 4y + 4z = 7

-2x + 3y - z = 1

-4x + 2y - 4z = -4

Answer 1

The same variable must be eliminated from each pair of equations.

 

First, Pair up the 1st and 2nd equations:

 

  -2x + 4y + 4z = 7

  -2x  + 3y - z   = 1

 

Variable X can be eliminated by subtraction:

 

     y + 5z = 6   <--- Call this equation ALPHA

 

Now pair up the 2nd and 3rd equations:

 

-2X + 3y - z = 1

-4x + 2y - 4z = -4

 

Remember we have to eliminate the same variable, so 

we must eliminate variable X. This can be done by

multiplying the 2nd equation by -2, then adding.

 

4x + -6y + 2z = -2

4x + 2y  - 4z = -4

 

-4y - 2z = -6

4y + 2z = 6 <--- multiplies everything by -1; call this equation BETA

 

 

ALPHA:  y + 5z = 6

 

BETA:   4y+ 2z = 6

 

Multiplies equation ALPHA by -4, then add them together.

 

  -4y + -20z = -24

   4y +    2z =   6

 

    -18z = -18

     z = 1

 

Substituting z=1 into equation ALPHA:

  y + 5z = 6

  y + 5(1) = 6

   y + 5 = 6

   y  = 1

 

Substituting y=z=1 into the original second equation:

 

   -2x + 3y - z = 1

   -2X + 3(1) - 1 = 1

   -2X +3 - 1 = 1

    -2X + 2 = 1

     -2X = -1

        X = 1/2

 

 So x=1/2 and y=z=1

 

The check is left for you.