System of Linear Equations Calculator (2 Variables)
System of 2 Linear Equations Calculator system of two equations
Enter the six coefficients of a₁x + b₁y = c₁ and a₂x + b₂y = c₂. The tool applies Cramer's Rule, then cross-checks with elimination and substitution.
The calculator above solves a system of two linear equations with Cramer's rule and explains the same result with substitution and elimination.
Formulas used
a₁x + b₁y = c₁, a₂x + b₂y = c₂
D = a₁b₂ − a₂b₁, Dx = c₁b₂ − c₂b₁, Dy = a₁c₂ − a₂c₁
x = Dx/D, y = Dy/D
- D ≠ 0: one unique solution (the two lines intersect).
- D = 0, Dx or Dy ≠ 0: no solution (parallel lines).
- D = Dx = Dy = 0: infinitely many solutions (the lines coincide).
Worked example
Solve x + y = 5 and x − y = 1. Here D = −2, Dx = −6, Dy = −4, so:
x = 3, y = 2
Enter the same values in the calculator above to see every step of the solution.
How to use the calculator
- Enter the coefficients a₁, b₁, c₁ and a₂, b₂, c₂.
- Press the calculate button. The result appears instantly.
- Read the step-by-step solution, rendered with KaTeX, to check your own work.