System of Equations Solver
Solve two or more equations at once and see the determinant steps and where the lines intersect.
Press Enter to solve · ^ for powers · sqrt() for roots · separate a system with a comma
The system of equations solver finds the values that satisfy every equation at the same time. Separate the equations with a comma, for example x + y = 10, x - y = 2, and the solver returns each variable with the working and a graph of both lines.
How to use the system of equations solver
- Type each equation, separated by a comma or semicolon.
- Use any letters you like — x and y, or a, b and c.
- Press Enter to see Cramer’s rule for 2×2 linear systems, or the full solution for larger systems.
Three ways to solve a system
Substitution
Solve one equation for a variable and substitute it into the other. Best when a variable already has a coefficient of 1.
Elimination
Add or subtract the equations to cancel one variable. Best when coefficients line up or are easy to match.
Cramer’s rule
Use determinants: x = Dₓ / D and y = Dᵧ / D. It is quick for 2×2 systems and is the method the solver shows.
Example: x + y = 10 and x − y = 2
- D = (1)(−1) − (1)(1) = −2.
- Dₓ = (10)(−1) − (2)(1) = −12, Dᵧ = (1)(2) − (1)(10) = −8.
- x = −12 / −2 = 6, y = −8 / −2 = 4.
Questions about systems of equations
What does the solution of a system represent?
For two linear equations, the solution is the point where the two lines cross. Every equation in the system is true at that point.
Can it solve three equations with three unknowns?
Yes. Enter all three equations separated by commas, for example x + y + z = 6, x − y = 0, 2x + z = 5.
Is substitution or elimination better?
Neither is always better. Use substitution when one variable is easy to isolate, and elimination when matching coefficients is quicker.