Solving Systems of Equations (3 Variables)

Solving Systems of Equations (3 Variables)

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

Example 1

Solve the system of equations. x + y + z = 3, 2x - y + z = 6, x + 2y - z = -4.

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 2x - y + z = 6.

Then x - 2y = 3.

System of linear equations: substitution method

Put z = -x - y + 3
into x + 2y - z = -4.

Then 2x + 3y = -1.

System of linear equations: substitution method

Now there are 2 variables (x, y)
and 2 equations. (x - 2y = 3, 2x + 3y = -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

If x, y, and z satisfy the below conditions, find the value of x + y + z. x + 4y = 5, y + 4z = 7, z + 4x = 8.

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

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