Solving systems of three linear equations in three variables

Solving systems of three linear equations in three variables

Systems of three linear equations in three variables 3x3

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

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