Solving systems of two linear equations in two variables

With this calculator you can easily solve systems of two linear equations in two variables.
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2 x 2 System of linear equations solver

Solving systems of two linear equations in two variables online

Free online calculator to solve systems of linear equations in 2 variables

The system of two linear equations with two unknowns has the form:

a11x1 + a12x2 = b1
a21x1 + a22x2 = b2

where x1 and x2 are the unknowns, a11, ..., a22 are the coefficients of the system and b1 and b2 are the constant terms.

To solve a system of two equations with two unknowns using the online calculator, simply enter the coefficients of the system and press 'Solve'.

There are several methods for solving a 2x2 system of linear equations.

Solving 2x2 systems of linear equations using Cramer's rule

Consider the application of Cramer's rule to solve the system of two equations with 2 variables.

Solving a system of two equations with 2 variables

As the system below has 2 rows and 2 columns and its determinant is non-zero, Cramer's rule may be applied.

Cramer's rule to solve a system of two equations

Solving systems of 2 linear equations in 2 variables by Gaussian Elimination

In the Gaussian Elimination method you eliminate variables by transforming the system of equations into row-echelon form by means of row operations. Then the system is solved by back-substitution.

Let's solve the following system of equations using Gaussian elimination

system of two linear equations

Divide the first equation by 3

solve system of two linear equations by Gaussian Elimination
Multiply (*) by 4 and add -1 times to the second equation. We get the following system

solve system of equations using Gaussian Elimination

From the second equation y=2. Substitution this into the first equation gives x.

solve system of equations

x=4, y=2.

Solving a system of two linear equations in two variables with substitution method

The main idea of the substitution method is to solve one of the variables in terms of the others (it does not matter which equation we choose) and then substitute the result into another equation.

Consider the following system:

system of two linear equations

Solving the second equation for y gives y = 4x - 14, and substituting this into the first equation gives x = (16-2(4x-14))/3.

Solving the first equation we get x = 4.

Substituting x = 4 into the second equation gives y = 2.

Thursday, March 23, 2017

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