How to use it
Choose a system of two or three equations, then enter the coefficients one row at a time. Write each equation in standard form first, with the variables on the left in the order x, y, z and the constant on the right. For example, \(y = 3 - 2x\) becomes \(2x + y = 3\). If a variable is missing from an equation, leave its box empty or enter 0.
Cramer's rule
The tool finds the determinant \(D\) of the coefficient matrix. For a 2×2 system that's the product down the main diagonal minus the product of the other diagonal; for 3×3 it uses the diagonal method. Then, for each variable, it swaps that variable's column for the column of constants and finds a new determinant, \(D_x\), \(D_y\) and, for three equations, \(D_z\). The solution is
\[x = \frac{D_x}{D}, \qquad y = \frac{D_y}{D}, \qquad z = \frac{D_z}{D}.\]
All the arithmetic uses exact fractions, so you'll see \(\frac{7}{3}\), not 2.333.
When D is zero
If \(D = 0\), Cramer's rule doesn't apply, and the system is one of two special cases. It's inconsistent (no solution) when the equations contradict each other, as \(x + y = 2\) and \(x + y = 5\) do. It's dependent (infinitely many solutions) when one equation is just a multiple or combination of the others, like \(x + y = 2\) and \(2x + 2y = 4\). The tool tells you which case you have.
With two variables you can picture it: each equation is a line, and the solution is the point where the lines cross. Parallel lines never meet, so there's no solution; two equations for the same line share every point.