Systems of Three Equations

Solving by Addition and Subtraction

Addition/Subtraction discussed how to solve systems of two equations with two variables by the Addition/Subtraction method. Systems with three equations and three variables can also be solved using the Addition/Subtraction method.

Pick any two pairs of equations in the system. Then use addition and subtraction to eliminate the same variable from both pairs of equations. This leaves two equations with two variables--one equation from each pair. Solve this system using the Addition/Subtraction method. Then plug the solution back in to one of the original three equations to solve for the remaining variable.

Here, in step format, is how to solve a system with three equations and three variables:

1. Pick any two pairs of equations from the system.
2. Eliminate the same variable from each pair using the Addition/Subtraction method.
3. Solve the system of the two new equations using the Addition/Subtraction method.
4. Substitute the solution back into one of the original equations and solve for the third variable.
5. Check by plugging the solution into one of the other three equations.

Example: Solve the following system:

4x - 3y + z = - 10
2x + y + 3z = 0
- x + 2y - 5z = 17

1. Pick two pairs:
   

   4x - 3y + z = - 10

   2x + y + 3z = 0

   and
2. 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

3. Solve the system of the two new equations:
   

   -5y - 5z = - 10
   5y - 7z = 34
   
   -12z = 24
   Thus, z = - 2
   
   -5y - 5(- 2) = - 10
   -5y = - 20
   Thus, y = 4

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
   Therefore, (x, y, z) = (1, 4, - 2).
5. Check: Does 2(1) + 4 + 3(- 2) = 0? Yes.