## Solving systems of three linear equations in three variables

#### Systems of three linear equations in three variables 3x3

a

_{11}x

_{1} + a

_{12}x

_{2} + a

_{13}x

_{3} = b

_{1}
a

_{21}x

_{1} + a

_{22}x

_{2} + a

_{23}x

_{3} = b

_{2}
a

_{31}x

_{1} + a

_{32}x

_{2} + a

_{33}x

_{3} = b

_{3}
where x

_{1}, x

_{2}, x

_{3} are the unknowns, a

_{11},..., a

_{33} are the coefficients of the system, b

_{1}, b

_{2}, b

_{3} are the constant terms

### 3x3 system of linear equations solver

System solver can be used for solving systems of three linear equations in three variables or checking the solutions of 3 x 3 systems of linear equations solved by hand.

To solve a system of three linear equations with three unknowns using the 3x3 system of equations solver, enter the coefficients of the three linear equations and click 'Solve'.

#### Solving a system of three linear equations in three variables using Cramer's rule

Example. Solving the system of three linear equations in three variables using

Cramer's rule.

By Cramer's rule:

#### Solving systems of three equations using Gaussian Elimination

Solving a system of linear equations using Gaussian Elimination
Example. Solving the system of three linear equations in three variables using Gaussian Elimination.

Divide the first equation by 3

Multiply (**) by 4 and add -1 times to the second equation, then multiply (**) by (-1) and add to the third equation. We get the following system:

Divide the second equation by

and get

Multiply (***) by

and add -1 times to the third equation.

The system we get

From the third equation z=3. Substitute this to the second equation:

=> y=1>

Substituting y and z to the first equation, we get x

=> x=5

x=5, y=1, z=3