Math
System of Equations Calculator (2×2)
Find x and y in seconds and see whether the system has a solution
This system of equations calculator solves 2×2 linear systems of the form ax + by = c: it gives you x and y, the determinants and whether the system has one solution, none or infinitely many.
The key idea
Solving a system is not just isolating a variable: it is knowing whether a solution exists
If the determinant D = a₁b₂ − a₂b₁ is 0, the two lines are parallel or the same line, and there is no single solution. In 2x + 3y = 12 and x − y = 1, D = −5, so there is exactly one solution: x = 3, y = 2. With x + y = 2 and 2x + 2y = 5, D = 0 and the system has no solution.
How it is calculated
Cramer's rule uses three determinants:
D = a₁b₂ − a₂b₁ Dx = c₁b₂ − c₂b₁ Dy = a₁c₂ − a₂c₁
x = Dx / D y = Dy / D
If D = 0, the system has no single solution: either it has none (parallel lines) or it has infinitely many (the same line).
Worked example
2x + 3y = 12 and x − y = 1: D = −2 − 3 = −5, Dx = −12 − 3 = −15, Dy = 2 − 12 = −10.
x = 3, y = 2
Common questions
Frequently asked questions
Does the method I use to solve a system matter?
No: substitution, elimination, graphing and Cramer's rule all lead to the same solution. Only the path changes, not the answer; Cramer's rule is the most direct for a 2×2 system.
How do I check the solution?
Plug x and y into both equations: both must hold. With x = 3 and y = 2: 2 · 3 + 3 · 2 = 12 and 3 − 2 = 1.
What is a consistent and independent system?
It is a system with exactly one solution: the two lines cross at one point. It happens when the determinant D = a₁b₂ − a₂b₁ is not 0.
When is a system inconsistent or dependent?
When the determinant D is 0. If Dx and Dy are also 0, it is dependent (infinitely many solutions, the same line); otherwise, it is inconsistent (parallel lines, no solution).
How do I enter an equation like y = 2x − 1?
Rewrite it in the form ax + by = c first. y = 2x − 1 becomes −2x + y = −1, so a = −2, b = 1 and c = −1.
Does it solve systems of three equations?
No, this calculator solves systems of two equations in two unknowns. For a 3×3 system, Cramer's rule works the same way, but with third-order determinants.
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