*25 points! * state in detailed terms on how to solve a three variable system, complete with an example.

Pick 2 pairs of equations then use addition and subtraction to eliminate the same variable from both pairs of equations then it is left with 2 variables 
Pick two pairs
4x - 3y + z = - 102x + y + 3z = 0
eliminate the same variable from each system
4x - 3y + z = - 10 
2x + y + 3z = 0 

4x - 3y + z = - 10 
-4x - 2y - 6z = 0 

-5y - 5z = - 10 

2x + y + 3z = 0 
- x + 2y - 5z = 17 

2x + y + 3z = 0 
-2x + 4y - 10z = 34 

5y - 7z = 34
Solve the system of the two new equations:
-5y - 5z = - 10 
5y - 7z = 34 

-12z = 24 
which is , z = - 2 

-5y - 5(- 2) = - 10 
-5y = - 20 
wich is , y = 4
substitute into one of the original equations
- x + 2y - 5z = 17 
- x + 2(4) - 5(- 2) = 17 
- x + 18 = 17 
- x = - 1 
x = 1 
which is , (x, y, z) = (1, 4, - 2) 
Does 2(1) + 4 + 3(- 2) = 0 ? Yes