Two unknowns
2x + 3y = 13 and x − y = −1 give x = 2, y = 3.
System of Linear Equations Solver
Solve systems like 2x + 3y = 13 and x − y = −1, written the normal way.
One per line, like 2x + 3y = 13. Terms can be on both sides.
Solution
Solution
x = 2, y = 3
Runs in your browser — nothing you enter is sent to a server.
How the System of Linear Equations Solver works
Write each equation on its own line the way you would on paper, such as 2x + 3y = 13. Terms can be on both sides, numbers can be decimals, and variables can be any letters. The solver moves everything to one side and uses Gauss–Jordan elimination.
If the equations contradict each other it says there is no solution; if some repeat others (say one is another times 2) it says there are infinitely many.
Formula
[A | b] → row operations → [I | x]
The coefficients form a matrix A and the constants a column b; row operations turn A into the identity, leaving the solution.
Examples
Three unknowns
x + y + z = 6, 2y + 5z = −4 and 2x + 5y − z = 27 give x = 5, y = 3, z = −2.
Frequently asked questions
What does “no solution” mean?
The lines (or planes) never meet — for example x + y = 2 and x + y = 3 are parallel.
Can I write 3a = 2b + 1?
Yes. Terms on the right are moved across automatically.
How many equations can I solve?
Up to 8 equations with up to 8 unknowns.
Related tools
Matrix Calculator
Add, subtract and multiply matrices, and find the determinant, inverse and transpose.
Math Tools
Quadratic Equation Solver
Solve ax² + bx + c = 0 with real or complex roots, the discriminant and the vertex, with working.
Math Tools
Fraction Calculator
Add, subtract, multiply and divide fractions, with the answer simplified and shown as a mixed number and decimal.
Math Tools