Systems of Equations Solver

Solve two or three simultaneous equations by substitution or elimination, with every step shown.

Try an example:

What a system of equations is

A system is a set of equations that must all be true at the same time. Its solution is the set of values — an $(x, y)$ pair when there are two unknowns — that satisfies every equation at once. On a graph, it's the point where the lines cross.

How to use the systems of equations solver

Type or paste your problem into the box above — for example 2x + 3y = 12, x - y = 1. Use ^ for powers, / for fractions and sqrt() for square roots, or describe it in words.
Press Enter. The solver works through the problem with every step shown; switch Steps off if you only want the final answer.
Ask a follow-up in the same chat — "why did you do that?" or "now check it" — and it answers with the earlier working in context.

Three possible outcomes

One solution: the lines cross at a single point.
No solution: the lines are parallel, and the algebra ends in a contradiction such as $0 = 5$ .
Infinitely many solutions: both equations describe the same line, and the algebra collapses to $0 = 0$ .

Substitution or elimination?

Use substitution when one equation already has a variable on its own, like $y = 2 x - 1$ : put that expression into the other equation.
Use elimination when coefficients line up: add or subtract the equations, scaling first if needed, so one variable cancels.
Solve for the remaining variable, substitute back to find the other, and check the pair in both original equations.

Worked example: $2 x + 3 y = 12$ and $x - y = 1$

Isolate a variableFrom the second equation, $x = y + 1$ .
Substitute $2 (y + 1) + 3 y = 12$ , so $5 y + 2 = 12$ .
Solve for $y$ $5 y = 10$ , so $y = 2$ .
Back-substitute $x = 2 + 1 = 3$ .
Check $2 (3) + 3 (2) = 12$ and $3 - 2 = 1$ . Both hold, so the solution is $(3, 2)$ .

Common mistakes to avoid

Substituting into the same equation

If you rearranged the second equation, substitute into the first. Using the same one just gives $0 = 0$ .

Scaling only part of an equation

When you multiply an equation to line up coefficients, multiply every term — including the constant on the right.

Stopping after one variable

Finding $y = 2$ is half the answer. The solution is a pair, and it should be checked in both equations.

Related calculators

Browse all calculators

Questions students ask

Can it solve three equations with three unknowns?

Yes. Separate the equations with commas or new lines. It eliminates one variable to reduce the system to two equations, then solves those.

Can I choose the method?

Yes. Add by elimination or by substitution to your question. You can also ask for the matrix method or Cramer's rule.

Does it solve nonlinear systems?

Yes, such as a line and a parabola: y = x + 1, y = x^2 - 1. Systems like this can have more than one solution, and the steps find all of them.

How does it show there's no solution?

The steps reach a false statement like $0 = 7$ and explain that the lines are parallel, so they never meet.