System of Equations Calculator

Cramer's rule expresses each unknown 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 det = a1 b2 - a2 b1. Then x = (c1 b2 - c2 b1) / det and y = (a1 c2 - a2 c1) / det. The 3x3 version replaces each variable's column with the constants column and divides by the same main determinant. This calculator uses the rule directly so the steps match what you would do by hand.

A determinant of 0 means Cramer's rule cannot return a single answer because the coefficient matrix is singular. Two things can happen. If the equations are consistent (every replaced determinant is also 0), there are infinite solutions and the lines or planes overlap. If they are inconsistent (some replaced determinant is non-zero), there is no solution and the lines are parallel or the planes never meet. The calculator runs that consistency check for you and labels the result.

Cramer's rule is fastest for 2x2 and 3x3 systems when you only need the values of x, y, and z. Substitution works well when one equation is already solved for a variable or has a coefficient of 1. Elimination (row reduction) scales better to large systems and shows you whether the system is dependent or inconsistent. All three methods give the same answer when the system has a unique solution.

Anywhere two or more quantities are linked by linear relationships. Common examples: balancing a budget with multiple income and expense streams, mixing two solutions to hit a target concentration, balancing chemical reactions, solving a circuit with Kirchhoff's laws, pricing tickets when you know totals and revenue, and fitting a line or plane through data points.

Yes. Enter any real number, including decimals, fractions written as decimals (1/3 as 0.333...), and negative values. The calculator rounds the final solution to six decimal places to hide floating-point noise but keeps full precision internally during the calculation.