Question 72918
<pre>
Solve the following:

3x-5y+2z=19
5x+2y-3z=-8
-2x+3y+5z=7

To solve for systems of 3 linear equation in 3 variable, we use Substitution
Method, Elimination Method, Gaussian Reduction method, Cramer's rules.

I choose Elimination Method

equation 1 :      3x-5y+2z=19
equation 2 :      5x+2y-3z=-8
equation 3 :      -2x+3y+5z=7


Let us Eliminate x in equation 1 and equation 2.

              3x-5y+2z=19
              5x+2y-3z=-8
             _____________

Since we cannot eliminate 3x and 5x by adding, we will find a certain
number to that will be multiplied to the two equations.

Multiply 5 to 3x-5y+2z=19 and -3 to 5x+2y-3z=-8


               5(3x-5y+2z=19)
              -3(5x+2y-3z=-8)
             _________________

             15x - 25y +10z = 95
            -15x -  6y + 9z = 24    Add
            ____________________

                 - 31y + 19z = 119      Equation 4


Now eliminate x in equation 1 and equation 3. Multiply 2 to 3x-5y+2z=19
and 3 to -2x+3y+5z=7

             2(3x-5y+2z=19)
             3(-2x+3y+5z=7)
             ______________

             6x - 10y +  4z = 38
            -6x +  9y + 15z = 21
            ____________________


                   -y + 19z = 59   Equation 5


Then using equation 4 and equation 5 eliminate another variable.
I choose z then solve for y.

              -31y + 19z = 119      
          -1(  - y + 19z = 59  )
          ______________________


             -31y + 19z = 119  
                y - 19z = -59  
             ________________
              -30y     = 60    
                     y = -2

The value of y will be substituted to either equation 5 or 4
I choose equation 5

               -y + 19z = 59, y = -2
            -(-2) + 19z = 59  
                    19z = 59 - 2
                    19z = 57          Divide 19 both sides
                      z = 3


The Substitute y = -2, z = 3 to equation 1 or 2 or 3.
I choose equation 3.

                -2x + 3y + 5z = 7, y = -2, z = 3
           -2x + 3(-2) + 5(3) = 7
                 -2x - 6 + 15 = 7
                          -2x = -2
                            x = 1


Checking: 
equation 1 :      3x-5y+2z=19, x = 1, y = -2, z = 3
                  3(1)-5(-2)+2(3)=19
                      3 + 10 + 6 = 19
                              19 = 19  ---------->True


equation 2 :      5x+2y-3z=-8, x = 1, y = -2, z = 3
                  5(1)+2(-2)-3(3)=-8
                       5 - 4 - 9 = -8
                              -8 = -8 ------------>True


equation 3 :       -2x+3y+5z=7, x = 1, y = -2, z = 3
                  -2(1)+3(-2)+5(3)= 7
                       -2 -6 + 15 = 7
                                7 = 7 ----------> True


Therefore, the solution is x = 1, y = -2 and z = 3