2×2 Systems of Equations Solver

Solve a₁x + b₁y = c₁; a₂x + b₂y = c₂. Cramer's rule via determinants.

Inputs

Result

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General calculation reads

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How Not to Be Wrong
Jordan Ellenberg - math thinking
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The Joy of X
Steven Strogatz - accessible math
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Scientific calculator (Casio fx-991EX)
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How to use this calculator

  • Enter coefficients of both equations.
  • Read x and y.

About this calculator

Cramer's rule for 2×2 systems: x = Dx/D, y = Dy/D where D = a₁b₂ − a₂b₁ is the coefficient determinant. When D = 0, the lines are parallel (no solution) or identical (infinite solutions). For larger systems (3×3, 4×4) Cramer's rule still works but Gaussian elimination is faster. 2×2 is the bread-and-butter case in physics (mixture problems), economics (supply/demand), and elementary algebra.

Frequently asked

When does D = 0?+
When the two equation lines are parallel (different y-intercepts: no solution) or identical (same line: infinite solutions).
Why Cramer's rule for 2×2?+
Closed-form, fast, and intuitive. For 3+ variables, Gaussian elimination scales better.
Is this exact?+
For rational inputs, yes. JavaScript Number gives 15 sig figs.
Substitution method?+
Solve one equation for x, plug into the other. Same answer; just a different process.
How is determinant computed?+
For 2×2 matrix [[a, b], [c, d]]: ad − bc. Geometric meaning: signed area of the parallelogram spanned by the rows.

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