Calxsolver
Algebra

System of Equations (2×2)

This solver handles a system of two linear equations, a₁x + b₁y = c₁ and a₂x + b₂y = c₂. It computes the coefficient determinant D, then x and y from the ratios Dₓ/D and Dᵧ/D. When D = 0 the lines are parallel or identical and no unique solution exists.

System of Equations (2×2)

Solve two linear equations in x and y by Cramer's rule.

Try:
Answerx = 2, y = 1
  1. Equation 12x + 1y = 5
  2. Equation 21x − 1y = 1
  3. DeterminantD = a₁b₂ − a₂b₁ = 2·-1 − 1·1 = -3
  4. Dₓc₁b₂ − c₂b₁ = -6
  5. Dᵧa₁c₂ − a₂c₁ = -3
  6. xDₓ / D = -6 / -3 = 2
  7. yDᵧ / D = -3 / -3 = 1
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Frequently asked questions

What method is used?

Cramer's rule: the unknowns are ratios of determinants, x = Dₓ/D and y = Dᵧ/D.

What does D = 0 mean?

A zero determinant means the two lines are parallel (no solution) or identical (infinitely many) — there is no single solution.

How do I enter my equations?

Write each equation in the form ax + by = c and enter the six coefficients a₁, b₁, c₁, a₂, b₂, c₂.

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