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System of Equations Solver
Use this system of equations solver to solve two or three linear equations at the same time. Enter the coefficients, and the solver works through Gaussian elimination step by step, shows the exact solution, and tells you clearly when a system has no solution or infinitely many solutions instead of guessing.
Enter the numbers in front of each unknown and the number on the right of the equals sign. A missing term is 0; a term with no number in front of it is 1.
How it works
A system of equations is a set of equations that must all be true together, using the same unknowns. One equation with two unknowns has many answers; two equations with two unknowns usually pin the answer down to exactly one pair. The solver uses Gaussian elimination, the standard method taught in algebra classes: it rearranges the system without changing its solution, until the answer can be read off directly.
The coefficients are the numbers in front of each unknown. In x + y = 5, the coefficient of x is 1 and the coefficient of y is 1, and the constant on the right is 5. In 2x − y = 1, the coefficients are 2 and −1, and the constant is 1. Type each number into its own box, including the minus sign for negative numbers. Leave a box as 0 when that unknown does not appear in an equation.
Gaussian elimination works in two phases. First, eliminate: use the first equation to remove x from the second equation (and, for a 3×3 system, from the third too), then use the new second equation to remove y from the third. What is left is a triangular system where the last equation has only one unknown. Second, back-substitute: solve that last equation, plug the answer into the equation above it, solve that one, and keep going up until every unknown is known. The steps box on the page shows each elimination and each substitution, so you can follow the working and use it in your own homework.
No solution happens when elimination produces a false statement such as 0 = 4. That means the equations contradict each other — for two equations, the lines are parallel and never meet. Infinitely many solutions happens when elimination produces a true statement such as 0 = 0. That means one equation is a multiple of another, so the lines are the same line and every point on it works. The solver reports both cases in plain words and explains what it found, instead of dividing by zero and showing an error.
One boundary is worth stating clearly. The existing equation solver page on this site solves single equations — one quadratic, cubic, or linear equation in one unknown. This page solves systems of equations only: two or three equations that must hold at the same time. If you have one equation and one unknown, use that page; if you have a system, you are in the right place.
Worked example
Solve the system x + y = 5 and x − y = 1. This is the classic first system, and the diagram above shows it: two lines crossing at one point.
Step 1: Write the system down. Equation 1 is 1x + 1y = 5. Equation 2 is 1x − 1y = 1.
Step 2: Add the two equations to eliminate y. The y terms cancel: (1 + 1)x = 5 + 1, so 2x = 6.
Step 3: Divide by 2. x = 3.
Step 4: Substitute x = 3 back into the first equation. 3 + y = 5, so y = 2.
Step 5: Check in the second equation. 3 − 2 = 1, which is correct.
Try it yourself: choose "2 equations", enter 1, 1, 5 in the first row and 1, −1, 1 in the second row, and press Solve. The tool gives exactly the same answer — x = 3, y = 2 — with the elimination and substitution steps shown. For a 3×3 check, try x + y + z = 6, x − y + z = 2, and 2x + y − z = 1: the solver gives x = 1, y = 2, z = 3.
FAQ
What is a system of equations?
A system of equations is two or more equations that must all be true at the same time, using the same unknowns. For example, x + y = 5 and x - y = 1 form a system in x and y. The solution, x = 3 and y = 2, is the one pair of values that satisfies both equations at once.
How does Gaussian elimination work?
Gaussian elimination simplifies a system step by step: use the first equation to eliminate the first unknown from the equations below it, then use the second equation to eliminate the second unknown from the ones below that, until the system is triangular. Then work backwards from the last equation to find each unknown in turn. This solver shows every one of those steps.
What does it mean when a system has no solution?
It means no values can satisfy all the equations at once — the equations contradict each other. For example, x + y = 3 and x + y = 5 cannot both be true, because x + y cannot equal two different numbers. Geometrically, the lines are parallel and never meet. The solver reports this clearly instead of giving a wrong answer.
What does it mean when a system has infinitely many solutions?
It means the equations are really the same equation in disguise, so every point on one line also lies on the other. For example, x + y = 3 and 2x + 2y = 6 are the same line, and any pair with x + y = 3 works — (0, 3), (1, 2), (3, 0), and so on. The solver says so and shows the relationship between the unknowns.
What is the difference between this solver and an equation solver?
This page solves systems of several equations with several unknowns at the same time, such as 2x2 and 3x3 linear systems. The equation solver on this site solves a single equation in one unknown — a quadratic, cubic, or linear equation on its own. If you have one equation and one unknown, use that page; if you have two or three equations to satisfy together, use this one.