Systems of Equations Calculator
Solves a₁x + b₁y = c₁ and a₂x + b₂y = c₂ with Cramer’s rule, reporting unique, none, or infinitely many solutions plus determinant steps.
Limitation: This worksheet solves exactly two linear equations in two unknowns.
How to Solve a 2×2 System
Write both equations as a x + b y = c, then enter the six coefficients.
Rewrite each equation in standard form a x + b y = c.
Move constants to the right
If an equation is written as 2x + y − 8 = 0, treat c as 8 so the form stays a x + b y = c.
Interpreting the Solution Status
A nonzero determinant yields a unique intersection point. A zero determinant means the lines are parallel: either the same line (infinite solutions) or distinct parallels (no solution).
Worked System Examples
Unique intersection
Coefficients
Solution
D = −3, Dx = −9, Dy = −6 → x = 3, y = 2.
No solution
Coefficients
Solution
D = 0 while Dx ≠ 0, so the equations contradict.
Check by substitution
Plug the reported x and y back into both original equations whenever D is small.
Frequently Asked Questions
Still have questions about this calculation?
Try the CalculatorCramer’s Rule for 2×2 Systems
Cramer’s rule expresses the solution with determinants when the coefficient matrix is invertible.
Formula
Coefficient determinant
D = a₁b₂ − a₂b₁
Numerator determinants
Dx = c₁b₂ − c₂b₁; Dy = a₁c₂ − a₂c₁
Unique solution
x = Dx/D; y = Dy/D
Scientific Background
Assumptions: both equations are linear and already in standard form. Limitations: floating-point rounding can blur near-singular systems; exact classroom fractions may need symbolic work.
Sources & review
Systems of Equations Calculator documents the references behind its formulas and assumptions. Always treat results as educational estimates, not personalized professional advice.
Last reviewed
July 22, 2026
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