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Certified calculator · Version 1

System of Equations Solver

Solve two linear equations in x and y using a certified deterministic 2×2 system engine.

Solution (x, y)(3, 2)
x3
y2
Determinant-2
CalcuMint Pro

Advanced Analysis

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Advanced chartsExplore the result across time and assumptions.
Scenario comparisonCompare multiple assumptions side by side.
Sensitivity analysisSee which assumptions change the outcome most.
Goal SolverSolve backwards from the outcome you want.
Result methodology

The certified engine solves the two-equation system from the same validated coefficients using its non-zero determinant.

Solution steps
  1. D = ae - bd = -2
  2. x = (ce - bf) / D = 3
  3. y = (af - cd) / D = 2

How this calculator works

For ax + by = c and dx + ey = f, use the non-zero determinant ae - bd to compute x and y.

Formula

x=(ce-bf)/(ae-bd), y=(af-cd)/(ae-bd)

Solves ax + by = c and dx + ey = f using the non-zero 2×2 determinant.

Worked example

x+y=5; x-y=1 is a verified test case used by the calculation engine.

Assumptions and limitations

  • The two equations must have one unique solution.
  • Inputs are finite real numbers.

Methodology & sources

This calculator uses deterministic, versioned calculation logic. The formula and verified examples above are part of the calculation definition used by CalcuMint.

  • CalcuMint deterministic algebra engine — Standard two-variable linear-system identities.

Frequently asked questions

What systems are supported?

This version solves two linear equations in two variables with a unique solution.

What happens for dependent or inconsistent equations?

A zero determinant is rejected because the system does not have one unique solution.

Related calculators

Linear Equation SolverSolve ax + b = 0 with deterministic algebra steps.Matrix Inverse CalculatorInvert a non-singular 2×2 matrix with visible steps.