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System of Linear Equations Solver

Solve 2×2, 3×3 and 4×4 linear systems by Gaussian elimination, Cramer's rule or the matrix inverse — with the determinant, step-by-step working, and detection of no-solution and infinite-solution cases.

Related tools

All Algebra & equations tools →
System of Nonlinear Equations SolverSolve 2- or 3-variable nonlinear systems (polynomial, trigonometric, exponential) with multivariate Newton's method — enter equations as text, set a starting guess, and surface every real solution with its residual.Linear Equation SolverSolve linear equations like 2x + 3 = 7, x/2 + 4 = 7 or 7x − 5 = 3x + 11 in one variable, with exact fractions and step-by-step working.Linear Programming SolverSolve linear programs with the simplex method: maximize or minimize a linear objective under ≤, ≥ and = constraints with up to 8 variables, returning the optimal value, the optimal point and detecting infeasible or unbounded problems.System of ODEs SolverSolves 2×2 and 3×3 linear systems x′ = Ax with eigenvalues, eigenvectors and the closed-form general solution, classifies the fixed point (node, spiral, centre, saddle, degenerate), and integrates arbitrary nonlinear 2D systems numerically with a phase portrait.System of Inequalities GrapherGraph a system of linear inequalities in x and y: shade the feasible region, draw each boundary (dashed for strict), list the region's vertices and tell you whether it is bounded, unbounded or empty.Absolute Value Equation SolverSolve absolute-value equations like |x+3| = 5, 3|x+2| = 12 or |x+1| = |x−3| with step-by-step case analysis and exact solutions.Absolute Value Inequality SolverSolve absolute-value inequalities such as |x+3| < 5 or |x−2| > 3 and get the solution as an interval, with AND/OR reasoning explained step by step.Cubic Equation SolverSolve ax³ + bx² + cx + d = 0 for all three roots (real and complex), with the discriminant, the nature of the roots and steps.

Open System of Linear Equations Solver and you get an answer straight away, with no account to create. It sits under Algebra & equations in our catalogue, alongside System of Nonlinear Equations Solver and Linear Equation Solver.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What does System of Linear Equations Solver do?

Solve 2×2, 3×3 and 4×4 linear systems by Gaussian elimination, Cramer's rule or the matrix inverse — with the determinant, step-by-step working, and detection of no-solution and infinite-solution cases.

When would I actually use this?

Getting through a problem set: solving for the unknown, factoring an expression, and — more usefully — seeing the steps that got there.

What is the most common mistake?

Losing a solution when both sides are squared or divided by an expression. Squaring can add roots that do not satisfy the original equation, and dividing can remove one — check every answer back in the original.

How is System of Linear Equations Solver different from System of Nonlinear Equations Solver?

They sit next to each other but answer different questions: System of Nonlinear Equations Solver is the one to open when you need it to solve 2- or 3-variable nonlinear systems (polynomial, trigonometric, exponential) with multivariate Newton's method — enter equations as text, set a starting guess, and surface every real solution with its residual. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

Linear Equation Solver is the closest one after this: Solve linear equations like 2x + 3 = 7, x/2 + 4 = 7 or 7x − 5 = 3x + 11 in one variable, with exact fractions and step-by-step working.

What else is worth having open alongside it?

Linear Programming Solver and System of ODEs Solver — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

The algebra is symbolic where the expression allows it and numeric otherwise, and the tool says which it used. Exact results are given as fractions or radicals rather than rounded decimals.

Further reading

All guides →
How-toHow to Use the Quadratic Formula: Solving ax² + bx + c = 0Solve any quadratic equation with x = (−b ± √(b² − 4ac)) / 2a, and read the discriminant to know how many real solutions to expect.How-toHow to Find a Cube Root by HandA step-by-step guide to finding a cube root by estimation and refinement — plus how it differs from a square root and when a value stays irrational.ExplainerHow Exponents Work: Powers and the Laws of ExponentsAn exponent is repeated multiplication. Learn the laws of exponents, what negative and fractional powers mean, and why anything to the zero is one.GuideThe Four Kinematics Equations: Which One to Use, and What Each One Leaves OutFive variables, four equations, and each equation is missing exactly one of them. Choose by looking at the variable the question never mentions.ExplainerThe Flat Pattern of a Cone Is a Sector, Not a CircleRoll a 50 mm cone 80 mm tall and its pattern is a 190.8° slice of a 94 mm circle. Flatten a nearly cylindrical lampshade and the pattern radius runs to 839 mm — the shallower the taper, the further away the apex.ExplainerA 1 % Drop Rate Does Not Mean One Hundred RunsAt 1 %, a hundred attempts give you 63.4 % — not certainty. Ninety per cent takes 230 attempts and ninety-nine takes 459, and more than a third of players are still empty-handed at a hundred.
System of Linear Equations Solver — OneKitly