🟰Simultaneous Equations Solver

About this tool

Type one equation per line as you'd write it, such as 2x + 3y = 13 and x - y = -1. Terms can be on either side, coefficients can be decimals or fractions such as 1/2x, and any single letters work as variables. The solver gives exact answers, checks them against every equation, lists the elimination steps, and tells you when there's no solution or infinitely many.

How simultaneous equations are solved

Simultaneous equations are several equations that must all be true at once. With two unknowns, each linear equation is a straight line, and the solution is where the lines cross. You can solve them by substitution, rearranging one equation for one variable and substituting it into the other, or by elimination, adding or subtracting multiples of the equations to cancel a variable. This tool uses elimination in a systematic form called Gaussian elimination, working with exact fractions so there are no rounding errors: 2x + 3y = 13 and x − y = −1 give x = 2 and y = 3. Three equations in three unknowns describe three planes, which usually meet at a single point; x + y + z = 6, 2x − y + z = 3 and x + 2y − z = 2 give x = 1, y = 2, z = 3. Two things can go wrong. If the equations contradict each other, such as x + y = 2 and x + y = 3, the lines are parallel and there's no solution. If one equation is just a multiple of another, such as x + y = 2 and 2x + 2y = 4, they describe the same line and there are infinitely many solutions. Honest limits: this solves linear equations only, so terms like x², xy or 1/x aren't allowed, and up to four unknowns. The matrix calculator, the quadratic solver and the graphing calculator are related.

Frequently asked questions