System of equations calculator (2 unknowns)
Solve a system of two linear equations (ax + by = c) with Cramer's rule.
How the calculation works
A 2×2 linear system has two equations of the form a·x + b·y = c and looks for the (x, y) that satisfies both — geometrically, where the two lines cross.
The calculator uses Cramer's rule. First the determinant:
D = a₁·b₂ − a₂·b₁
- If D ≠ 0: unique solution →
x = (c₁·b₂ − c₂·b₁)/D,y = (a₁·c₂ − a₂·c₁)/D - If D = 0: the lines are parallel (no solution) or coincident (infinite).
Step-by-step example
2x + y = 5 and x − y = 1:
- D = −3
- x = 2, y = 1
Examples
- x + y = 10 and x − y = 2: solution x = 6, y = 4.
- 2x + 2y = 4 and x + y = 2: D = 0, coincident lines → infinite solutions.
Frequently asked questions
What if the equations aren't in ax + by = c form?
Rearrange first: move the unknowns to the left and the number to the right.
What does "no solution" mean?
The two lines are parallel and never meet — no (x, y) satisfies both.
Does Cramer work for 3 unknowns?
Yes, with 3×3 determinants. This calculator handles the 2×2 case, the most common in school.
What does a zero determinant mean?
There is no unique solution: the system is either indeterminate (coincident lines) or impossible (parallel lines).
Which method is best?
Substitution when a variable is already isolated, elimination when coefficients are neat, Cramer when you want a mechanical route.
How do I check my answer?
Substitute the values into BOTH original equations, not just the last one you used.
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Updated on June 30, 2026 · by Rafael Rossi · Methodology & sources