Linear Equations Solver
Solve systems of 2 or 3 linear equations with step-by-step working — Cramer's rule and row reduction. Handles unique, no, and infinite solutions. Free.
System of Linear Equations Solver
How to Use the Linear Equations Solver
Choose the system size
Pick two equations (two unknowns, x and y) or three equations (three unknowns, x, y, z).
Enter the coefficients
Type the number in front of each variable and the constant on the right of each equation. Leave a box blank for a coefficient of zero.
Read the solution
The values of x, y (and z) appear instantly, computed with Cramer's rule, alongside the row-reduced form of the system so you can see the method.
Understand special cases
If there is no unique answer, the solver tells you whether the system has no solution (inconsistent) or infinitely many (dependent), rather than just failing.
Solving Simultaneous Equations
Where Two Lines Meet
A system of linear equations asks a simple geometric question: where do these lines (or planes) meet? Two equations in x and y each describe a straight line, and solving the system means finding the single point (x, y) that lies on both — the intersection. Three equations in x, y, and z each describe a plane in space, and the solution is the point where all three planes cross. This is one of the most useful procedures in all of mathematics because so many real problems reduce to it: mixing solutions to a target concentration, balancing supply and demand, fitting a curve through points, distributing a budget across constraints, or solving a circuit. The two classical hand methods are substitution (solve one equation for a variable and plug it into the others) and elimination (add or subtract multiples of equations to cancel a variable). This solver uses the matrix equivalents — Cramer's rule and row reduction — which scale cleanly and are exactly how computers solve such systems.
The engine behind the answer is the determinant. Every square system has a coefficient determinant, a single number computed from the coefficients, and it acts as a gatekeeper: if it is non-zero, the lines or planes meet at exactly one point and there is a unique solution, which Cramer's rule delivers by computing a determinant for each variable. If the determinant is zero, something special is happening — the equations are not independent. The solver then looks more carefully, using row reduction, to tell you which of two cases you are in: the equations might be contradictory (parallel lines that never meet, giving no solution at all), or they might secretly be the same constraint written differently (giving infinitely many solutions). Most calculators just say "error" when the determinant is zero; this one explains what the zero means, which is the part that actually teaches you something.
"Solving a system is finding where lines or planes intersect. The determinant is the gatekeeper: non-zero means one clean answer; zero means the equations are hiding something."
From Classroom to Computation
Systems of equations are a cornerstone of secondary and early-university mathematics, and they are the doorway to linear algebra — the branch of maths that underpins computer graphics, machine learning, economics, and engineering. The reason the subject scales so well is that everything generalises: the same row-reduction method that solves a 2×2 by hand solves a 2,000×2,000 system inside a physics simulation, and the same determinant that decides a unique solution here decides whether a matrix can be inverted there. This tool focuses on the 2×2 and 3×3 cases that students meet most, showing both the clean answer and the row-reduced form so the method is visible, not hidden. Because it runs entirely in your browser, it is an instant, private way to check homework, settle a calculation, or just see how a system resolves — with an honest, explained answer even in the awkward cases where no single solution exists.
10 Facts About Linear Systems
Solving a 2-equation system finds where two lines intersect.
A 3-equation system finds where three planes meet in space.
The two hand methods are substitution and elimination.
Cramer's rule solves a system using determinants.
A non-zero determinant means exactly one solution.
A zero determinant means no unique solution — none or infinitely many.
Parallel lines (inconsistent) give no solution.
Identical equations (dependent) give infinitely many solutions.
Row reduction (RREF) is how computers solve large systems.
Linear systems are the doorway to linear algebra and machine learning.
Frequently Asked Questions
- Enter each equation's coefficients and constant, and the solver finds the values of the unknowns that satisfy all the equations at once. It uses Cramer's rule (with determinants) and shows the row-reduced form so you can follow the method. Choose 2 equations for x and y, or 3 for x, y, and z.
- Cramer's rule solves a square system using determinants: each variable equals the determinant of the coefficient matrix with that variable's column replaced by the constants, divided by the main coefficient determinant. It works whenever that main determinant is non-zero — that is, whenever the system has exactly one solution.
- It happens when the coefficient determinant is zero, and there are two cases. If the equations contradict each other (like parallel lines), there is no solution. If they are really the same equation in disguise (dependent), there are infinitely many solutions. This solver checks which case applies and tells you, instead of just showing an error.
- Yes — switch to the "3 equations" mode for systems in x, y, and z. The solver handles the larger determinant and row reduction automatically and reports the solution or the special case, just as it does for 2×2 systems.
- It is the system after Gauss-Jordan elimination, where each equation has been simplified so the solution can be read directly. The vertical line separates the coefficients from the constants. Seeing the RREF makes the elimination method concrete and is exactly how computers solve large systems.
- Type the actual number. For a coefficient of 1, enter 1; for a missing variable, enter 0 or leave the box blank (it is treated as zero). For a negative coefficient, include the minus sign. Decimals and fractions written as decimals are fine.
- Solutions are shown as decimals, rounded for readability. Whole-number answers appear exactly; fractional answers appear as their decimal value. This keeps the output clear for systems with any coefficients.
- Everywhere two or more conditions must hold at once: mixing problems, supply and demand, circuit analysis, curve fitting, budgeting, and the linear algebra behind computer graphics and machine learning. The same method that solves a 2×2 here scales to the huge systems inside simulations.
- Yes. The solver runs in your browser — nothing is uploaded, stored, or logged — and it works offline once loaded.
- Completely free, with no account, sign-up, or limit. It runs entirely in your browser and collects no data. Use it as often as you like.
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