Cramer's Rule Calculator (2x2)

This calculator solves a system of two linear equations in two unknowns using Cramer's rule, a method built on determinants rather than substitution or elimination. Enter the six coefficients for your two equations, written as a1x + b1y = c1 and a2x + b2y = c2, then the calculator works out x, y and the main determinant D. Behind the scenes it forms D from a1b2 minus b1a2, then swaps in the constants c1 and c2 to build two more determinants, one for each unknown, dividing each by D to get the answer. The result card shows x as the headline figure, with y and the determinant D shown alongside it, so you can check your working or confirm an exam answer in seconds. If D comes out as zero, the calculator flags that the system has no unique solution, meaning the two lines are parallel or identical rather than crossing at one point. Use it to check homework, verify simultaneous equations by hand, or quickly solve a real system that turns up in physics, economics or engineering work. A worked example below shows the full calculation step by step so you can follow the method rather than just reading off a number, and the formula and assumptions sections explain when Cramer's rule applies and where it loses its edge over other methods for larger systems.

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Determinant D

The formula

With main determinant D = a1 b2 minus b1 a2, the solution is x = (c1 b2 minus b1 c2) / D and y = (a1 c2 minus c1 a2) / D. If D is zero the system has either no solution or infinitely many.

Worked example

For 2x + y = 5 and x minus y = 1: D = 2(minus 1) minus 1(1) = minus 3, x = (5(minus 1) minus 1(1))/(minus 3) = 2, y = (2(1) minus 5(1))/(minus 3) = 1. Enter the six coefficients to confirm.

Frequently asked questions

What is Cramer's rule?

A method that solves a linear system using ratios of determinants, one for each unknown.

When does it work?

When the coefficient determinant D is non-zero, giving exactly one solution. If D is zero, the system has no unique solution.

Does it scale to bigger systems?

Yes, in principle, but it becomes slow for large systems, where elimination methods are preferred.

Who this calculator is for

This calculator is for algebra students and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid numbers.
The standard method is applied.
Results are rounded for display.

Formula and sources

Related calculators

Quadratic Formula Calculator: solve a quadratic.
Scientific Calculator: general maths.
Matrix Calculator: matrix operations.

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