System of Linear Equations Solver
Solve 2×2 and 3×3 systems of linear equations instantly with our free linear equations solver. Uses Gaussian elimination with partial pivoting — see every row operation as an augmented matrix, with automatic solution verification. No signup required, all calculations run locally in your browser.
Enter the coefficients of your system, select 2×2 or 3×3, and click Solve. The system is solved using Gaussian elimination with full step-by-step matrix operations. All calculations run instantly in your browser — no data is ever sent to any server.
Enter Coefficients — each row is one equation
Quick Examples
Why Use Our System of Linear Equations Solver?
Instant System of Equations Solving
Solve 2×2 and 3×3 systems of linear equations in milliseconds using Gaussian elimination. Our linear equations solver handles any consistent, inconsistent, or dependent system — all processed instantly in your browser.
Secure Linear Equations Solver Online
All system of equations calculations run 100% locally in your browser. Your coefficients never leave your device — use our linear equations solver online with complete privacy and zero data collection.
Linear Equations Solver — No Installation
Use our system of linear equations solver directly in any browser with no downloads, plugins, or app installs required. Solve 2×2 and 3×3 systems from any device, anywhere, with no setup needed.
Gaussian Elimination with Matrix Steps
Every solution includes a full step-by-step Gaussian elimination walkthrough — row swaps, row operations, and back substitution — displayed as augmented matrices. Results are verified against all original equations.
Common Use Cases for System of Linear Equations Solver
Algebra & Linear Algebra Education
Students solving systems of equations for homework or exam preparation can use our solver to check their work and understand the Gaussian elimination method step by step. The augmented matrix display makes the process transparent.
Engineering & Physics Problems
Kirchhoff's voltage and current laws, structural analysis, and fluid dynamics all produce systems of linear equations. Our linear equations solver handles 2×2 and 3×3 systems with exact fractional results.
Economics & Business Modelling
Supply and demand equilibrium, cost-revenue analysis, and portfolio optimisation often reduce to systems of linear equations. Our solver gives exact solutions with verification against all original equations.
Programming & Algorithm Verification
Developers implementing linear algebra libraries or numerical methods can use our system of linear equations solver to verify their Gaussian elimination implementations against known correct outputs.
Chemistry & Balancing Equations
Balancing chemical equations and stoichiometry problems can be formulated as systems of linear equations. Our solver handles the coefficient matrix directly, making chemical equation balancing straightforward.
Computer Graphics & Geometry
Finding intersection points of lines and planes, transforming coordinates, and solving geometric constraints all involve systems of linear equations. Our 2×2 and 3×3 solver covers the most common geometric cases.
Understanding Systems of Linear Equations
What is a System of Linear Equations?
A system of linear equations is a collection of two or more linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. A 2×2 system has two equations and two unknowns (x, y); a 3×3 system has three equations and three unknowns (x, y, z). Systems can have exactly one solution (consistent and independent), infinitely many solutions (consistent and dependent), or no solution (inconsistent). Our linear equations solver uses Gaussian elimination to determine which case applies and find the solution when it exists.
How Our Linear Equations Solver Works
- 1. Enter coefficients: Fill in the coefficient grid — each row is one equation. Enter the coefficient of each variable and the right-hand side value. Select 2×2 or 3×3 system size.
- 2. Instant solving: Click Solve. The system is solved using Gaussian elimination with partial pivoting for numerical stability — all processed locally in your browser.
- 3. View steps and verify: Click "Show Steps" to see the full augmented matrix at each elimination step. The solution is automatically verified against all original equations.
Gaussian Elimination — How It Works
- Step 1: Form the Augmented Matrix: Write the system as an augmented matrix [A|b] where A contains the coefficients and b contains the right-hand side values. For example, 2x − y = 3 and x + 3y = 7 becomes [[2, −1, | 3], [1, 3, | 7]].
- Step 2: Forward Elimination: Use row operations to create zeros below each pivot element, transforming the matrix into upper triangular form (row echelon form). Partial pivoting (swapping rows to use the largest available pivot) improves numerical stability.
- Step 3: Back Substitution: Starting from the last equation (which has only one unknown), solve for each variable in reverse order. Substitute known values back into earlier equations to find all unknowns.
Three Types of Solutions
- Unique solution: The system has exactly one solution. Geometrically, two lines intersect at one point (2×2) or three planes meet at one point (3×3).
- No solution (inconsistent): The equations contradict each other. Geometrically, the lines are parallel (2×2) or the planes don't share a common point (3×3).
- Infinite solutions (dependent): The equations are redundant — one is a multiple of another. Geometrically, the lines coincide (2×2) or the planes share a line (3×3).
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Frequently Asked Questions About System of Linear Equations Solver
A system of linear equations solver finds the values of variables that satisfy all equations in the system simultaneously. Our solver uses Gaussian elimination to solve 2×2 and 3×3 systems, shows step-by-step matrix operations, and verifies the solution — all processed instantly in your browser with no signup required.
Gaussian elimination is a systematic method for solving systems of linear equations by transforming the augmented matrix into upper triangular form (row echelon form) using row operations, then solving for each variable using back substitution. It is the standard algorithm used in linear algebra and numerical computing.
A system has no solution when the equations are inconsistent — they contradict each other. For example, x + y = 3 and x + y = 5 cannot both be true. Geometrically, the lines (2×2) or planes (3×3) are parallel and never intersect. Gaussian elimination detects this when a row becomes [0, 0, …, 0 | c] where c ≠ 0.
A system has infinitely many solutions when the equations are dependent — one equation is a linear combination of the others. For example, x + y = 3 and 2x + 2y = 6 are the same equation. Geometrically, the lines coincide. Gaussian elimination detects this when a row becomes all zeros.
Yes. Our linear equations solver accepts any real number as a coefficient, including decimals (e.g. 1.5, -0.25) and integers. Results are displayed as exact fractions where possible (e.g. 2/3 instead of 0.666667) for maximum precision.
Partial pivoting is a technique in Gaussian elimination where rows are swapped to place the largest available coefficient in the pivot position. This improves numerical stability and prevents division by very small numbers that could cause large rounding errors. Our solver uses partial pivoting automatically.
Yes, completely. All calculations run 100% locally in your browser using JavaScript. Your coefficients are never sent to any server, stored, or logged. Use our system of linear equations solver online with full confidence in your privacy.
Yes. Our system of linear equations solver is 100% free with no signup, no account, no premium tier, and no usage limits. Solve as many systems as you need — completely free, forever.
For a 2×2 system ax + by = e and cx + dy = f, you can use substitution (solve one equation for x, substitute into the other) or elimination (multiply equations to cancel one variable). The determinant method gives x = (ed − bf)/(ad − bc) and y = (af − ec)/(ad − bc). Our solver shows the Gaussian elimination approach step by step.