# Solving systems of linear equations in 3 variables

## Systems of three linear equations in three variables

a11x1 + a12x2 + a13x3 = b1
a21x1 + a22x2 + a23x3 = b2
a31x1 + a32x2 + a33x3 = b3

where x1, x2, x3 are the unknowns, a11,..., a33 are the coefficients of the system, b1, b2, b3 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

Thursday, April 24, 2014