Systems of Equations Calculator

Solve two or more equations together with substitution, elimination, or row operations.

Drag & drop or upload an image or PDF

Enter equations, such as 2x + y = 7, x - y = 2

Try an example

Find values that satisfy every equation

A system is solved only when the same variable values make all of its equations true. For two linear equations, the solution is the point where their lines intersect.

Substitution is convenient when one variable is already isolated. Elimination is often faster when coefficients line up. Larger systems can be organized with an augmented matrix and row operations.

How to use the systems of equations calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

A two-variable linear system

The pair (x, y) must satisfy both equations at the same time.

{a_{1} x + b_{1} y = c_{1} a_{2} x + b_{2} y = c_{2}

A reliable way to work through it

Substitution

Solve one equation for a variable and replace that variable in the other equation.

Elimination

Add or subtract equivalent equations so one variable cancels.

Row reduction

Use legal row operations on an augmented matrix, especially for systems with three or more variables.

Worked example

Solve 2x + y = 7 and x - y = 2

3 x = 9

Add the equations to eliminate y.

x = 3

Divide by 3.

3 - y = 2

Substitute x = 3 into the second equation.

y = 1

Solve the remaining one-variable equation.

The system has one solution: (3, 1).

Common mistakes to check

Changing only part of an equation

When multiplying an equation for elimination, every term must be multiplied.

Stopping after one variable

Substitute back to find the remaining variables and report an ordered pair or tuple.

Assuming every system has one solution

Parallel lines give no solution, while equivalent equations give infinitely many solutions.

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Questions students ask

How should I enter a system of equations?

Separate equations with a comma or a new line. Include an equals sign in each equation.

Can the calculator solve three equations?

Yes. Enter all three equations and variables. The solver can use elimination or matrix row reduction.

How does it show no solution?

The steps will end in a contradiction, such as 0 = 4, which means the equations are inconsistent.

Can it solve nonlinear systems?

Yes, although nonlinear systems may have several solutions and may require substitution, factoring, or numerical methods.