SOLUTION: Solve the system of equations. How do I solve one of the equations for one variable then substitute the solution back into the other 2 equations? <pre> 4x - 4y - z = 38 x +

Algebra ->  Test -> SOLUTION: Solve the system of equations. How do I solve one of the equations for one variable then substitute the solution back into the other 2 equations? <pre> 4x - 4y - z = 38 x +       Log On


   

Question 360587: Solve the system of equations. How do I solve one of the equations for one variable then substitute the solution back into the other 2 equations?
 4x - 4y -  z = 38
  x + 4y - 3z =  6
-3x + 2y -  z = -14

Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
(1)    4x - 4y -  z =  38
(2)     x + 4y - 3z =   6
(3)   -3x + 2y -  z = -14

Pick any one of the three equations and pick one of the variables in it to solve for.

I'll pick equation (2) and pick x to solve for:

(2)     x + 4y - 3z = 6
(4)               x = 6 - 4y + 3z

Now substitute 6 - 4y + 3z for x in the other two equations:

Substituting 6 - 4y + 3z for x in equation (1) and simplifying:

(1)                4x - 4y - z = 38
       4(6 - 4y + 3z) - 4y - z = 38 
       24 - 16y + 12z - 4y - z = 38
                24 - 20y + 11z = 38
(5)                 -20y + 11z = 14

Substituting 6 - 4y + 3z for x in equation (3) and simplifying:

(3)   -3x + 2y -  z = -14


(1)               -3x + 2y - z = -14
      -3(6 - 4y + 3z) + 2y - z = -14 
       -18 + 12y - 9z + 2y - z = -14
               -18 + 14y - 10z = -14
(6)                  14y - 10z = 4

Now we put equations (5) and (6) together as a system of two
equations and 2 variables:

(5)                 -20y + 11z = 14
(6)                  14y - 10z =  4

It's too hard to do that by substitution, so I'll use elimination:

To eliminate z, multiply equation (5) through by 10, and equation (6) 
through by 11, and add the resulting equations term by term:

                  -200y + 110z = 140
                   154y - 110z =  44
                  ------------------ 
                   -46y        = 184
(7)                          y = -4

Substituting in equation (6)

(6)                  14y - 10z =  4
                  14(-4) - 10z =  4
                     -56 - 10z =  4
                          -10z = 60 
(8)                          z = -6


Now we substitute for y amd z from equations (7) and (8) into equation (4)

(4)               x = 6 - 4y + 3z
                  x = 6 - 4(-4) + 3(-6)
                  x = 6 + 16 - 18
                  x = 4
              
Edwin

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