2×2 Systems of Equations Solver
Solve a₁x + b₁y = c₁; a₂x + b₂y = c₂. Cramer's rule via determinants.
Result
Solution
x = 3.500000, y = 2.000000
D = -14.0000.
- a₁x + b₁y = c₁2x + 3y = 13
- a₂x + b₂y = c₂4x + -1y = 12
- D (det)-14.0000
- Dx-49.0000
- Dy-28.0000
- x = Dx / D3.50000000
- y = Dy / D2.00000000
- Verify eq 113.0000
- Verify eq 212.0000
Step-by-step
- D = a₁b₂ − a₂b₁ = -14.0000.
- Dx = c₁b₂ − c₂b₁ = -49.0000.
- Dy = a₁c₂ − a₂c₁ = -28.0000.
- x = Dx/D = 3.500000; y = Dy/D = 2.000000.
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 the two equation lines are parallel (different y-intercepts: no solution) or identical (same line: infinite solutions).
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