###
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:

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

*Example*: Solve the following system:

4*x* - 3*y* + *z* = - 10

2*x* + *y* + 3*z* = 0

- *x* + 2*y* - 5*z* = 17

- Pick two pairs:

4*x* - 3*y* + *z* = - 10 |

2*x* + *y* + 3*z* = 0 |

and
- Eliminate the same variable from each system:

4*x* - 3*y* + *z* = - 10

2*x* + *y* + 3*z* = 0

4*x* - 3*y* + *z* = - 10

-4*x* - 2*y* - 6*z* = 0

-5*y* - 5*z* = - 10

2*x* + *y* + 3*z* = 0

- *x* + 2*y* - 5*z* = 17

2*x* + *y* + 3*z* = 0

-2*x* + 4*y* - 10*z* = 34

5*y* - 7*z* = 34

- Solve the system of the two new equations:

-5*y* - 5*z* = - 10

5*y* - 7*z* = 34

-12*z* = 24

Thus, *z* = - 2

-5*y* - 5(- 2) = - 10

-5*y* = - 20

Thus, *y* = 4

- Substitute into one of the original equations:

- *x* + 2*y* - 5*z* = 17

- *x* + 2(4) - 5(- 2) = 17

- *x* + 18 = 17

- *x* = - 1

*x* = 1

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