Cramer\u2019s Rule Calculator (2x2)
This Cramer\u2019s rule calculator solves a system of two linear equations in two unknowns using determinants. Enter the coefficients of the system, written as a1 x plus b1 y equals c1 and a2 x plus b2 y equals c2, and the calculator returns x and y. Cramer\u2019s rule expresses each unknown as a ratio of determinants: the main determinant of the coefficients in the denominator, and a modified determinant in the numerator. It gives a clean closed form and works whenever the main determinant is non-zero, meaning the system has a single unique solution. The method, a worked example and the assumptions are below.
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\u2019s 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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