System of Equations Solver

Enter the system in the supported format, verify the equations, and use the matrix or line-form tools to inspect the same relationship from another representation.

Stable numericsSubstitute / reconstructIndependent checkLocal calculation

Enter a₁x + b₁y = c₁ and a₂x + b₂y = c₂.

a₁

b₁

c₁

a₂

b₂

c₂

x = 3, y = 1Unique intersection

-3Determinant

one solutionSystem type

Algebra round-trip cluster

Solve once, then verify the same object through another representation

Independent round trips and invariants expose sign, scaling, reconstruction and degeneracy mistakes that a single formatted answer can hide.

Quadratics Matrices Systems

All 28 algebra tools

Matrix InverseA↔A⁻¹ identity round trips MidpointEndpoint reflection round trip Linear EquationScale invariance + degeneracy classes DeterminantTranspose/row-operation invariants Quadratic VertexStandard ↔ vertex coefficient round trip Slope InterceptPoints ↔ line-form equivalence Polynomial DivisionQuotient/remainder reconstruction Standard FormSlope-intercept ↔ standard round trip

3×3 RREF solver & residual audit

The primary 2×2 solver stays first. Use this optional extension for three linear equations in x, y, z. Partial-pivot row reduction classifies one/no/infinite solutions and keeps a row-operation trace.

xyzRHSEq 1Eq 2Eq 3

—System type

—Solution

—Max residual

RREFElimination audit

Ready.

Transform, then verify

Equivalent forms should agree. Use discriminants, roots, factoring, vertex form, substitution, matrix checks, or the Remainder Theorem where appropriate to verify the result rather than relying on a single opaque output.

Determinant zero is not automatically “no solution”

The system check distinguishes unique, inconsistent, and dependent/redundant equations, including a zero equation such as 0 = 0.

Scope and precision

These are bounded browser calculators, not a universal computer algebra system. WebToolArc keeps exact algebraic identities visible where practical, uses stable floating-point algorithms for numeric work, distinguishes undefined/no-solution/infinite-solution cases, and shows an independent check after the main Calculate action.

Use this result with confidence

Classify the system before trusting a numeric solution

A linear system can have one solution, no solution, or infinitely many solutions. A zero determinant in the 2×2 case does not distinguish the last two by itself. Row reduction exposes inconsistent rows and free variables directly, so classification should accompany any reported values.

Use residuals as an independent check

After solving for x, y, and z, substitute the values back into every original equation and measure the largest residual. A small residual supports the computed solution; a large one can reveal entry errors or unstable arithmetic. Keep the original coefficients with the residual so the check is reproducible.

Partial pivoting improves numerical stability

Gaussian elimination should choose a strong available pivot rather than divide by a tiny coefficient when a better row exists. Row swaps do not change the solution set. The 3×3 audit uses partial-pivot elimination and records its row operations so the calculation can be reviewed instead of treated as a black box.

Scale and conditioning still matter

A system with coefficients that differ by many orders of magnitude can be sensitive even when elimination succeeds. If a result will drive engineering, finance, or scientific decisions, compare with another solver or higher-precision method and inspect the residual relative to the size of the original coefficients.