Cramer's Rule Calculator

How Cramer's Rule Calculator Works?

Cramer's Rule solves a system of three linear equations with three unknowns by comparing determinants. The calculator represents the system as:

The coefficient matrix is $A=\left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right).\; Its\; determinant,\; called\; the\; main\; determinant,\; is\; calculated\; by\; expansion\; along\; the\; first\; row:$

Three additional determinants are formed by replacing one column of the coefficient matrix with the constants column. Replacing the first, second, or third column produces ${\Delta }_x$, ${\Delta }_y$, and ${\Delta }_z$, respectively. The unknowns are then found from:

The calculator accepts each coefficient and constant as a decimal value from -1,000,000 to 1,000,000. For example, the default inputs represent 2x + y − z = 3, x + 3y + 2z = 13, and 3x + 2y + z = 10. They produce $\Delta =10$, ${\Delta }_x=8$, ${\Delta }_y=30$, and ${\Delta }_z=16$, giving $x=0.8$, $y=3$, and $z=1.6$.

Practical Use Cases

Students can use the calculator to verify hand calculations in linear algebra assignments, especially when practicing determinants, matrix notation, and Cramer's Rule for three-equation systems.

Engineers can solve small simultaneous-equation models involving three unknown design or operating variables and quickly inspect the main and replacement determinants.

Researchers can use it for compact mathematical models in which three measured or estimated variables must satisfy three linear relationships.

Educators can demonstrate how changing one coefficient or constant affects the determinant values and the resulting solutions without manually recomputing every expansion.

Data and operations analysts can apply it to small resource-allocation or balancing problems represented by three linear constraints, provided the relationships are linear and the coefficient matrix has a nonzero determinant.

Accuracy and Limitations

The calculator evaluates the entered decimal coefficients using determinant formulas and reports the resulting determinants and solutions, subject to normal floating-point rounding. Cramer's Rule produces a unique solution only when the main determinant $\Delta$ is nonzero; if $\Delta =0$, the system may have no solution or infinitely many solutions, and the ratios for $x$, $y$, and $z$ are not defined. Values near zero can also make results sensitive to small input changes. The calculator is limited to exactly three equations and three unknowns, accepts inputs between -1,000,000 and 1,000,000 with a 0.01 entry step, and does not replace a separate check for inconsistent or dependent equations.

Quick Examples