System of equations solver
Enter two, three or four linear equations, one per line. See the elimination steps, the solution, and, for two variables, the lines crossing on a graph.
Try:
One solution, none, or infinitely many?
Line 1
Line 2
One solution Different slopes, so the lines cross at exactly one point, which is the one solution.
- y = 1x + 1
- y = -0.5x + 4
How to solve a system by elimination
- Line the equations up with like variables in columns.
- Multiply one or both equations so one variable has equal (or opposite) coefficients.
- Subtract (or add) the equations. That variable disappears.
- Solve for the remaining variable.
- Substitute back into either original equation to find the other.
- Check the pair in both equations.
By substitution
Solve one equation for one variable, for example , and substitute that expression into the other equation: . Now there’s a single variable: , so .
By graphing
Each linear equation in two variables is a line; the solution is the intersection point. Graphing shows the answer at a glance, but it’s only exact when the crossing lands on grid points. Plot your own on the graphing calculator.
Frequently asked questions
What is a system of equations?
Two or more equations that share the same variables. A solution is a set of values that makes every equation true at the same time.
Elimination or substitution: which is better?
Substitution is quickest when one variable already has a coefficient of 1 or −1 (like y = 2x + 1). Elimination is usually cleaner otherwise, and it extends to three or more equations. Both always give the same answer.
How can a system have no solution?
If the equations describe parallel lines (same slope, different intercept), they never meet, so no pair (x, y) satisfies both. Elimination shows this as a false statement like 0 = 5.
Can it solve 3 equations with 3 unknowns?
Yes. Enter three equations separated by semicolons or new lines, and the solver uses Gaussian elimination, up to four equations in four unknowns.