2×2 Systems: Unique, None, or Infinite
Two straight lines in a plane can cross once, never, or lie on top of each other. This guide connects that geometry to the determinant checks used in a transparent 2×2 solver.
Standard Form Comes First
Rewrite each equation as a x + b y = c before comparing coefficients. Constants on the left belong on the right-hand side after you move them.
The Determinant Triage
Compute D = a₁b₂ − a₂b₁. If D ≠ 0, the coefficient matrix is invertible and exactly one (x, y) pair exists. If D = 0, inspect Dx and Dy: both zero means dependent equations (infinite solutions); otherwise the system is inconsistent (no solution).
Near-zero determinants
Classroom fractions can look nonzero after decimal rounding. When D is tiny, substitute the reported solution back into both originals.
Solve a 2×2 system
Enter six coefficients and review determinant steps:
Open Systems of Equations Calculator