Solving systems of linear equations in 3 variables

Solving systems of three linear equations with three variables online

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

system of linear equations

The determinant of the coefficient matrix

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:

Cramer's rule
where determinant is the determinant of the matrix formed by replacing the j column with the column of the constant terms

determinant

Solving the system of three equations with Cramer's rule

system of three equations

Cramer's rule to solve the system of three equations

By Cramer's rule:

Solve the system of three equations with Cramer's rule

Solving systems of three equations using Gaussian Elimination

system of three equations
Divide the first equation by 3

solve system of three equations

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:

Solve systems of three equations by Gaussian Elimination

Divide the second equation by and get Solve systems of equations with three unknowns by Gaussian Elimination

Multiply (***) by and add -1 times to the third equation.

The system we get Gaussian Elimination to solve systems of three equations

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

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