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System of Equations Calculator

Two straight lines on a plane either cross at one point, run on as the same line, or never meet — and those three geometries are the three answers a linear pair can have. Budgets split between two items, circuit currents meeting, supply meeting demand — all arrive as such pairs. This calculator solves any 2×2 system by Cramer's rule and names the case, rather than leaving a division by zero to explain itself.

Enter values

The first equation reads a₁x + b₁y = c₁

A zero determinant means the lines are parallel (no solution) or identical (infinitely many)

Results update as you type.

Results

Solution
x = 2.00, y = 1.00
Determinant (a₁b₂ − a₂b₁)
-3
Solution type
One solution

How is the System of Equations calculated?

Six number fields hold the two equations: a₁x + b₁y = c₁ and a₂x + b₂y = c₂. The calculator first evaluates the determinant a₁b₂ − a₂b₁ of the coefficient matrix. A non-zero determinant means one solution, found by Cramer's rule with two small determinants. A zero determinant means the equations are dependent: when the constants follow the same proportion the lines coincide and every point on them solves the pair, otherwise the lines are parallel and there is none.

Formula

For a₁x + b₁y = c₁ and a₂x + b₂y = c₂ the determinant is D = a₁b₂ − a₂b₁. When D is not zero, x = (c₁b₂ − c₂b₁) ÷ D and y = (a₁c₂ − a₂c₁) ÷ D by Cramer's rule. D equal to 0 means the coefficient rows are proportional; the pair then has infinitely many solutions if c₁a₂ = c₂a₁ and c₁b₂ = c₂b₁ too, and none if those fail.

Example

Solve 2x + y = 5 and x − y = 1. The determinant is 2×(−1) − 1×1 = −3, which is not zero, so one solution exists: x = (5×(−1) − 1×1) ÷ (−3) = 2 and y = (2×1 − 1×5) ÷ (−3) = 1. Double the first equation to 4x + 2y = 10 and the determinant collapses to 0 while both constants double too — the same line, so infinite solutions. Change it to 2x + y = 7 instead and the determinant stays 0, but now the lines are parallel: no solution at all.

Things to remember

The determinant is the diagnosis: non-zero means one crossing; zero asks whether the constants kept the same proportion. Real-world systems usually have a unique solution, since perfectly parallel constraints are a modelling accident. Beyond two equations the logic scales to matrices unchanged.

Frequently Asked Questions

What does a zero determinant mean for the solution?

That the two lines are parallel or identical, so either every point on the line solves the pair or none does. Check the constants: if each right-hand side follows the same ratio as its coefficients, as in 2x + y = 5 and 4x + 2y = 10, the equations describe one line.

How does Cramer's rule solve two equations?

It replaces one column of coefficients at a time with the constants and divides by the main determinant. With D = a₁b₂ − a₂b₁, the x value is (c₁b₂ − c₂b₁) ÷ D and the y value is (a₁c₂ − a₂c₁) ÷ D — two three-term products and a division each, which is exactly what the calculator runs.

Can a system of two equations have no solution?

Yes — when the lines are parallel, sharing a slope but different intercepts, such as 2x + y = 5 against 2x + y = 7. Subtracting would demand 5 = 7, which is false, so no pair of numbers satisfies both equations and the calculator says so directly.

What are infinite solutions in a linear system?

The case where both equations are the same line, so every point on it satisfies the pair: 2x + y = 5 and 4x + 2y = 10 are written differently but constrain nothing differently. Any x can be chosen freely, and the y value follows from either equation.

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