System of Linear Equations Calculator

Instantly solve systems of linear equations with 2 or 3 variables. Obtain exact fractions and decimal solutions with detailed steps.

System of Linear Equations Calculator

Solve for x and y

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About Systems of Linear Equations

What is a System of Linear Equations?

A system of linear equations consists of two or more linear equations that share the same set of variables. The goal is to find the values for these variables that satisfy all equations simultaneously. For example, a system with two variables, $x$ and $y$ , is typically written as:

\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}

Common Solving Methods

There are several standard mathematical methods to solve systems of linear equations:

Substitution Method: Solve one equation for one variable and substitute it into the other equation.
Elimination Method: Add or subtract equations to eliminate one variable.
Cramer's Rule: Use determinants to find the solution (applicable for systems with a unique solution).
Matrix Method: Use inverse matrices or Gaussian elimination.

Types of Solutions

Unique Solution

The lines intersect at exactly one point. The system is consistent and independent.

Infinite Solutions

The lines are identical (overlap completely). The system is consistent and dependent.

No Solution

The lines are parallel and never intersect. The system is inconsistent.

FAQ

Can this calculator solve systems with 3 variables?

Yes, simply switch to the "3 Variables" tab to solve systems of the form $ax + by + cz = d$ .

What if I get "No Solution"?

This means the equations represent parallel lines (or planes) that never intersect, so there is no set of values that satisfies all equations simultaneously.

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