System of Equations Solver

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Two-variable linear system solved by Cramer's rule

Quick Answer

System of Equations Solver: Two-variable linear system solved by Cramer's rule. System: a₁x + b₁y = c₁; a₂x + b₂y = c₂.

x +y =

x = 2.8, y = 0.8

Verify: 2×2.8 + 3×0.8 = 8.0000

x2.8

y0.8

D-5.00

TypeUnique

Cramer's rule

1D = a₁b₂ − a₂b₁ = 2×-1 − 1×3 = -5
2Dₓ = c₁b₂ − c₂b₁ = 8×-1 − 2×3 = -14
3Dᵧ = a₁c₂ − a₂c₁ = 2×2 − 1×8 = -4
4x = Dₓ/D = 2.8；y = Dᵧ/D = 0.8

💡 Geometric view: each equation is a line; the solution is their intersection. D≠0 → one crossing point; D=0 → parallel (no solution) or coincident (infinite). Cramer's rule generalizes to n equations but gets slow — beyond 3×3, Gaussian elimination wins.

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Formula & How It Works

Two-variable linear system (Cramer's rule)

System: a₁x + b₁y = c₁; a₂x + b₂y = c₂

D = a₁b₂ − a₂b₁; Dx = c₁b₂ − c₂b₁; Dy = a₁c₂ − a₂c₁

D ≠ 0: unique solution x = Dx/D, y = Dy/D

D = 0 with Dx = Dy = 0: infinite solutions; otherwise no solution

Example: 2x+3y=12, x−y=1 → D = −5, Dx = −15 → x=3; Dy = −10 → y=2.

FAQ

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Source & Updates

Standard mathematical formulas. All logic verified. Last updated: 2025-07

Last updated: 2026-08-04

Formulas and content on this page are reviewed by the YiJianSuan team. For reference only — official sources prevail.

Cite this page: YiJianSuan. System of Equations Solver[EB/OL]. https://www.yijiansuan.com/en/c/math/linear-system

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