System of Equations Calculator (2×2)
Solve a system of two linear equations, a₁x + b₁y = c₁ and a₂x + b₂y = c₂.
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System of Equations Calculator (2×2) by TheCalcTools
How the System of Equations Calculator (2×2) works
Write each equation in the form ax + by = c and enter its three numbers.
The calculator computes the determinant D = a₁b₂ − a₂b₁.
If D is not zero, Cramer’s rule gives x and y directly.
The formula
Worked example
Using the default values shown in the calculator:
| a₁ | 2 |
|---|---|
| b₁ | 1 |
| c₁ | 5 |
| a₂ | 1 |
| b₂ | -1 |
| c₂ | 1 |
- The system is 2x + y = 5 and x − y = 1.
- D = 2×(−1) − 1×1 = -3.00.
- x = (5×(−1) − 1×1) ÷ D = 2.00, and y = (2×1 − 1×5) ÷ D = 1.00.
Result:
| Determinant | -3.00 |
|---|---|
| x | 2.0000 |
| y | 1.0000 |
Frequently asked questions
What does a zero determinant mean?
The two lines have the same slope. They are either parallel (no solution) or the same line (infinitely many solutions).
Is this the only way to solve a system?
No. Substitution and elimination give the same answer. Cramer’s rule is convenient for a calculator because it is a direct formula.
Sources and further reading
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This calculator gives estimates for information only. Check important results independently before you rely on them. See our disclaimer.