# Cramer's Rule Calculator

This calculator will solve the system of linear equations of any kind, with steps shown, using Cramer's rule.

Related calculators: System of Equations Calculator, System of Linear Equations Calculator

Comma-separated, for example, x+2y=5,3x+5y=14.
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Solve $\begin{cases} x + 3 y = 8 \\ - 2 x + 5 y = 17 \end{cases}$ for $x$, $y$ using the Cramer's rule.

## Solution

Write down the augmented matrix: $\left[\begin{array}{cc|c}1 & 3 & 8\\-2 & 5 & 17\end{array}\right]$.

Calculate the main determinant (for steps, see determinant calculator): $D = \left|\begin{array}{cc}1 & 3\\-2 & 5\end{array}\right| = 11$.

Substitute the $x$-column with the RHS (for steps in calculating the determinant, see determinant calculator): $D_{x} = \left|\begin{array}{cc}8 & 3\\17 & 5\end{array}\right| = -11$.

Hence, $x = \frac{D_{x}}{D} = \frac{-11}{11} = -1$.

Substitute the $y$-column with the RHS (for steps in calculating the determinant, see determinant calculator): $D_{y} = \left|\begin{array}{cc}1 & 8\\-2 & 17\end{array}\right| = 33$.

Hence, $y = \frac{D_{y}}{D} = \frac{33}{11} = 3$.

$x = -1$, $y = 3$A