JLJosh Liljenquist

Algebra calculator

System of equations calculator

Solve two linear equations in x and y from their coefficients. The calculator checks the determinant, distinguishes a unique solution from inconsistent or dependent equations, and displays substitution checks.

By

Two-variable systemsSolution classificationSubstitution check

Enter known values

Enter the known values and select Calculate.

How to use this calculator

  1. Write each equation in ax + by = c form.
  2. Enter all six coefficients, using zero for a missing term.
  3. Calculate and inspect the determinant classification.
  4. Substitute the displayed x and y values back into both equations.

Formula and method

For a₁x + b₁y = c₁ and a₂x + b₂y = c₂, determinant D = a₁b₂ − a₂b₁. When D ≠ 0, x = (c₁b₂ − c₂b₁) ÷ D and y = (a₁c₂ − a₂c₁) ÷ D.

Worked example

For 2x + y = 7 and x − y = 2, D = −3. The calculated solution is x = 3 and y = 1, which satisfies both equations.

Assumptions and result interpretation

  • Coefficients are finite real numbers.
  • The tool solves two linear equations in two variables.
  • A zero determinant requires consistency checks before labeling the system.
  • Decimal display is rounded while solution arithmetic keeps full precision.

Understanding the result

A nonzero determinant means the lines meet once. A zero determinant can mean parallel inconsistent lines or the same line represented by proportional equations.

Common mistakes

  • Moving a term without changing its sign
  • Leaving a missing coefficient blank instead of using zero
  • Dividing by a zero determinant
  • Assuming parallel lines always represent the same equation

Methodology sources

Quadratic formula calculator

Browse all calculators.

This deterministic utility is a calculation aid. Confirm definitions, required precision, and any course, laboratory, professional, or policy requirements separately.