System of Equations Solver
Solve a 2×2 system of linear equations — two equations in x and y — with Cramer’s rule, and see whether it has one solution, none, or infinitely many.
Calculator
- Cramer’s rule
- Determinant classification
- Unique / none / infinite
- Both-equation check
Result & calculation receipt
- Method
- Cramer's rule (D ≠ 0 → unique solution)
- Formula
x = (c₁b₂ − c₂b₁)/D, y = (a₁c₂ − a₂c₁)/D, D = a₁b₂ − a₂b₁- Substitution
- x = 2, y = 1
- Steps
- Coefficient determinant D = a₁b₂ − a₂b₁ = 2·-1 − 1·1 = -3.
- Dx = c₁b₂ − c₂b₁ = -6, x = Dx/D = 2.
- Dy = a₁c₂ − a₂c₁ = -3, y = Dy/D = 1.
- Verification
- Eq1: 2·2 + 1·1 = 5 = 5; Eq2: 1 = 1.
- Exact result
- x = 2, y = 1
- Decimal result
- x: 2 · y: 1
- Engine
- math-solvers v1.4.0
How this calculator works
The calculator forms the coefficient determinant D = a₁b₂ − a₂b₁. If D is non-zero there is a unique solution, given by x = Dx/D and y = Dy/D where Dx and Dy replace a column with the constants. If D is zero, the lines are either identical (infinitely many solutions) or parallel (no solution), which it distinguishes by comparing the constant determinants.
Worked example
For 2x + y = 5 and x − y = 1, the determinant is 2(−1) − 1(1) = −3. Then x = (5·−1 − 1·1)/−3 = 2 and y = (2·1 − 1·5)/−3 = 1. Both equations check: 2(2) + 1 = 5 and 2 − 1 = 1. A system like x + 2y = 3 and 2x + 4y = 6 has determinant 0 and matching ratios, so it has infinitely many solutions.
Key insight
The determinant is the switch that decides a linear system’s fate: non-zero means the two lines cross at exactly one point; zero means they are the same line or never meet. Cramer’s rule turns that geometry into two quick fractions.
Frequently asked questions
What is Cramer’s rule?
Cramer’s rule solves a linear system using determinants: x and y are ratios of a modified determinant to the coefficient determinant D = a₁b₂ − a₂b₁. It works whenever D is non-zero.
When does a 2×2 system have no solution?
When the coefficient determinant is zero and the equations are inconsistent — the two lines are parallel and never intersect. The calculator detects this and reports no solution.
When are there infinitely many solutions?
When the determinant is zero and the two equations are proportional — they describe the same line. Every point on that line is a solution.
How is the solution checked?
The calculator substitutes x and y back into both original equations and confirms each holds, showing the check in the receipt.
Can I solve this with matrices instead?
Yes — a 2×2 system is Ax = b, solvable by the matrix inverse. The matrix calculator handles that form, and the two methods agree.
Methodology & verification
The system is solved with Cramer’s rule. The coefficient determinant D = a₁b₂ − a₂b₁ classifies the system: D ≠ 0 gives the unique solution x = (c₁b₂ − c₂b₁)/D, y = (a₁c₂ − a₂c₁)/D; D = 0 is split into infinitely many solutions (consistent, proportional equations) and no solution (inconsistent). Solutions are given exactly and as decimals, and verified by substitution into both equations.
Engine: math-solvers v1.4.0 · Last reviewed: · Every result is computed client-side and never sent to a server. See the verification methodology and the scientific calculator validation.