About the Linear Equations Solver
Linear Equations Solver finds the values of the unknowns in a set of simultaneous linear equations. Choose two, three or four unknowns, fill in the coefficient of each unknown and the right-hand side of each equation, and the answer appears at once. The tool works with exact fractions, so a result such as 1/3 is shown as 1/3 and not as 0.3333, and it also gives decimals. It tells you when a system has no solution or has infinitely many, shows the determinant, checks the answer by putting the values back into each equation and lists every step of the Gauss–Jordan elimination so you can follow the working for homework or revision. Coefficients can be integers, decimals or fractions. Everything runs in your browser.
Updated September 2026
How to use the Linear Equations Solver
- 1Choose how many unknowns you have: 2, 3 or 4.
- 2Type the coefficient of each unknown in each equation, and the number on the right of the equals sign.
- 3Use 0 where an unknown is missing from an equation. Fractions like 1/3 and decimals are fine.
- 4Read the solution, the decimals and the determinant.
- 5Open Show the working to see each elimination step, or copy the solution.
What a system of equations is
A system of linear equations is a set of equations that share the same unknowns. You need as many independent equations as unknowns to find one answer. Problems like mixing two solutions, working out prices from two shopping bills, balancing chemical equations and finding currents in a circuit all lead to systems of this kind.
With two unknowns you can picture each equation as a line, and the solution is the point where the lines cross. With three unknowns each equation is a plane, and the solution is the point where three planes meet.
The three kinds of answer
- One solution: the lines or planes meet at a single point. The determinant is not zero.
- No solution: the equations contradict one another, as with parallel lines.
- Infinitely many solutions: the equations repeat the same information, so many points fit.
- The tool names which case applies, and shows the working that led to it.
Following the working
Gauss–Jordan elimination works on a table of the coefficients. For each unknown in turn, the tool picks a row that contains it, divides that row so the coefficient becomes one and subtracts multiples of it from the other rows so that the unknown disappears from them. When it is done, each row says one unknown equals a number. If a row of zeros ends in a nonzero number, the system has no solution.
Open Show the working to see the table after every step. The substitution check under the answer puts your values back into each equation, which is a good habit for any exam.
Related tools
Matrix Calculator adds, multiplies and inverts matrices, Quadratic Equation Solver handles equations with squares, Fraction Calculator adds and simplifies fractions, Slope Calculator works with lines through two points and Triangle Calculator solves triangles from their sides and angles.
Features
- Systems of 2, 3 or 4 linear equations
- Exact fractions with decimal values as well
- Detects no solution and infinitely many solutions
- Determinant of the coefficient matrix
- Step-by-step Gauss–Jordan elimination
- Check by substituting the answer back
- Accepts integers, decimals and fractions
- Runs in your browser
Frequently asked questions
How do I solve a system of linear equations?
Common methods are substitution, elimination and matrices. This tool uses Gauss–Jordan elimination: it adds multiples of one equation to the others to remove unknowns one at a time until each unknown has its own value. The working shows each step.
What does no solution mean?
It means the equations contradict each other, such as x + y = 2 and x + y = 5. Graphically the lines are parallel and never meet, or in three unknowns the planes do not share a common point.
What does infinitely many solutions mean?
It means at least one equation gives no new information, for example x + y = 2 and 2x + 2y = 4. The equations describe the same line or plane, so many combinations of values work. The tool tells you how many independent equations there are.
Why is the determinant shown?
The determinant of the coefficient matrix is zero exactly when the system does not have a single unique solution. If it is not zero, there is exactly one solution. It is also a quick way to check work in a linear algebra course.
Can I enter fractions and decimals?
Yes. Type 1/3, 0.5 or −2.75 in any box. The tool turns decimals into exact fractions before it starts, so there is no rounding during the working, and it shows the final answer as a fraction and a decimal.
How do I write an equation such as 2x + 3y = 8?
Put 2 in the first box, 3 in the second and 8 in the box after the equals sign. For x − y = 1, put 1 and −1. If an unknown is missing, for example in x + z = 4, put 0 for the missing coefficient of y.
Does it solve nonlinear equations such as x² + y = 5?
No. It solves only linear equations, where each unknown appears to the first power and is not multiplied by another unknown. For quadratics, use Quadratic Equation Solver.