SOLUTION: x + 3y + z = 10 2x + y - z = 1 3x - 2y - 8z = -3 RE: 3x3 Systems - Methods to solve this by substitution

 x + 3y +  z = 10
2x +  y -  z =  1
3x - 2y - 8z = -3


Algorithm for solving any system of equation entirely by substitution:

1. Pick any equation
2. Pick any letter in that equation and solve for it.
3. Substitute the expression you get for that in each of the other equations and simplify
4. You now have a system of one less equation in one less unknown.
5. Go back to step 1 and continue until you have just one equation in one unknown,
6. Solve for that letters.
7. Use "back substitution" to find the values of the other unknowns.

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1. Pick any equation

I'll pick 2x + y - z = 1

2. Pick any letter in that equation and solve for it

I'll pick y.  

          y = 1 - 2x + z 

3. Substitute the expression you get for that in each of the other equations
   and simplify.

                    x + 3y + z = 10
         x + 3(1 - 2x + z) + z = 10
           x + 3 - 6x + 3z + z = 10
                      -5x + 4z = 7     
            
                  3x - 2y - 8z = -3
       3x - 2(1 - 2x + z) - 8z = -3
         3x - 2 + 4x - 2z - 8z = -3
                      7x - 10z = -1`

4. You now have a system of one less equation in one less unknown.

      We have this system of only 2 equation and only 2 letters: 

                    -5x +  4z =  7
                     7x - 10z = -1
 
5. Go back to step 1 and continue until you have just one equation in one unknown,

1. Pick any equation

I'll pick -5x + 4z = 7

2. Pick any letter in that equation and solve for it.

I'll pick z
         4z = 7 + 5x
          z =  + x

3. Substitute the expression you get for that in each of the other equations and simplify.
 
         7x - 10z = -1
  
  7x - 10( + x) = -1

    7x -  - x = -1

Clear of fractions by multiplying through by 4

   28x - 70 - 50x = -4 
             -22x = 66   
        
4. You now have a system of one less equation in one less unknown.

   Yes we have one equation in one letter x


5. Go back to step 1 and continue until you have just one equation in one unknown.

We don't have to go back because we have just one equation in 1 letter.

6. Solve for that letter.

             -22x = 66
                x = 
                x = -3 

7. Use "back substitution" to find the values of the other unknowns.

Substitute x = -3 in

         7x - 10z = -1
      7(-3) - 10z = -1
        -21 - 10z = -1
             -10z = 20
                z = -2

Substitute x = -3 and z = -2 in
   
       2x + y - z = 1
 2(-3) + y - (-2) = 1 
       -6 + y + 2 = 1
            y - 4 = 1
                y = 5


Solution (x,y,z) = (-3,5,-2) 

[Note that you could have switched over to the method of elimination
(addition or subtraction) at any point.  But since you specified
substitution, I used only substitution.

Edwin