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:

4x- 3y+z= - 10

2x+y+ 3z= 0

-x+ 2y- 5z= 17

- Pick two pairs:

4 *x*- 3*y*+*z*= - 102 *x*+*y*+ 3*z*= 0 - 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.

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