## Solving systems of three linear equations with three variables online

## Systems of three linear equations in three variables

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

This online caluclator can be used for solving systems of 3 linear equations or checking the solutions of 3 by 3 systems solved by hand.

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

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

Consider a system of n linear equations for n variables

The determinant of the coefficient matrix

Cramer's rule may only be applied for n x n systems of linear equations if the determinant of the coefficient matrix is non-zero.

By Cramer's rule:

where

is the determinant of the matrix formed by replacing the j column with the column of the constant terms

Solving the system of three equations with Cramer's rule

By Cramer's rule:

#### Solving systems of three equations 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