System of Equations Calculator

Cramer's rule expresses each variable as a ratio of two determinants. For a 2x2 system a1*x + b1*y = c1 and a2*x + b2*y = c2, the main determinant is D = a1*b2 - a2*b1. Then x = (c1*b2 - c2*b1) / D and y = (a1*c2 - a2*c1) / D. For a 3x3 system the same logic applies: replace the column of the variable you want with the constants column and divide by the 3x3 main determinant. This calculator applies those formulas directly and reports the solution along with the value of the main determinant.

A determinant of 0 means the coefficient matrix is singular and no unique solution exists. The calculator compares the rank of the coefficient matrix with the rank of the augmented matrix. Equal ranks below the number of variables mean infinitely many solutions. A larger augmented rank means the system is inconsistent and has no solution.

Cramer's rule is a good fit for 2x2 and 3x3 systems when you want a direct closed-form formula for each variable. Substitution is quick when one equation is already isolated for a variable or has a coefficient of 1 (for example, y = 3x + 2). Elimination scales better beyond 3x3 and makes it easy to spot dependent or inconsistent systems early. All three methods give the same unique solution when one exists.

Anywhere two or more quantities are tied together by linear constraints. Typical examples include splitting a $1,500 budget across two expense categories, mixing a 10% and a 30% saline solution to hit a 20% target, applying Kirchhoff's laws to find current in a circuit, setting ticket prices when you know the number sold and total revenue, and finding the intersection of supply and demand curves in economics.

Yes. You can enter any real number as a coefficient or constant, including decimals and negative values. Fractions like 1/3 should be entered as their decimal equivalent (0.333333), since the inputs are read as plain decimals. The calculator carries full floating-point precision throughout and rounds each final result to six decimal places to suppress floating-point noise.