System of Equations Solver

Solve a system of two linear equations with two unknowns (x and y) using Cramer's rule, with every step shown so you can see exactly how the answer was reached.

x = —
y = —

How to Solve a System of Equations

Enter the coefficients of two linear equations to see x, y, and the working.

1

Enter both equations

Write each equation in the form a·x + b·y = c, then fill in the coefficients a, b, c for equation 1 and d, e, f for equation 2. Negative numbers and decimals both work.

2

The calculator applies Cramer's rule

It computes the determinant det = a·e − b·d, then solves x = (c·e − b·f) ÷ det and y = (a·f − c·d) ÷ det. If det is 0, the lines are parallel and there's no single (x, y) solution.

3

Read the result and steps

The values of x and y appear at the top, with the determinant and every formula step listed below. Need to solve a single quadratic instead? The Quadratic Formula Calculator handles one-variable equations of degree two.

Frequently Asked Questions

What is a system of equations?
A system of equations is a set of two or more equations that share the same variables, where you're looking for values that satisfy every equation at once. This calculator handles a 2x2 linear system — two equations, two unknowns (x and y) — such as 2x + y = 5 and x − y = 1. Graphically, each linear equation is a straight line, and the solution is the point where the lines cross.
How do you solve a system of two linear equations?
This calculator uses Cramer's rule. For a·x + b·y = c and d·x + e·y = f, first compute the determinant det = a·e − b·d. Then x = (c·e − b·f) ÷ det and y = (a·f − c·d) ÷ det. Each of those three calculations is a straightforward multiply-and-subtract, which is why Cramer's rule is fast to apply by hand or by calculator for a 2x2 system.
What does it mean when a system has no unique solution?
When the determinant (a·e − b·d) equals 0, Cramer's rule can't divide by it, and the system has no single (x, y) answer. Geometrically this means the two lines are parallel: either they never cross (no solution at all) or they're the exact same line (infinitely many solutions). This calculator flags that case instead of returning a number.
What is Cramer's rule?
Cramer's rule is a formula for solving a system of linear equations using determinants instead of substitution or elimination. For a 2x2 system it needs just one determinant of the coefficient matrix (det = a·e − b·d) plus two more determinants swapped in for x and y, giving x = (c·e − b·f) ÷ det and y = (a·f − c·d) ÷ det directly. It scales to larger systems too, though substitution or elimination is often faster by hand once you're past 2 or 3 equations.