# Solving Systems of Equations (3 Variables)

How to solve systems of equations in 3 variables: examples and their solutions.

## Example 1

There are 3 variables (*x*, *y*, *z*) and 3 equations.

To solve this system,

use each equation to remove each variable.

To remove *z*,

choose one of the equation (*x* + *y* + *z* = 3)

and change it to '*z* = '.(*z* = -*x* - *y* + 3)

Put *z* = -*x* - *y* + 3

into 2*x* - *y* + *z* = 6.

Then *x* - 2*y* = 3.

System of linear equations: substitution method

Put *z* = -*x* - *y* + 3

into *x* + 2*y* - *z* = -4.

Then 2*x* + 3*y* = -1.

System of linear equations: substitution method

Now there are 2 variables (*x*, *y*)

and 2 equations. (*x* - 2*y* = 3, 2*x* + 3*y* = -1)

Solve this system.

Then *y* = -1 and *x* = 1.

System of linear equations: elimination method

To find *z*,

put *x* = 1 and *y* = -1

into *z* = -*x* - *y* + 3.

Then *z* = 3.

## Example 2

The numbers don't look good to solve.

Then try to find a pattern.

The coefficients are 1 and 4.

And your goal is to find *x* + *y* + *z*,

whose coefficients are all 1.

So add all three equations:

5*x* + 5*y* + 5*z* = 20.

And divide both sides by 5:*x* + *y* + *z* = 4.