# System of Linear Equations Calculator

This calculator will solve the system of linear equations of any kind, with steps shown, using either the Gauss-Jordan elimination method, the inverse matrix method, or Cramer's rule.

Related calculator: System of Equations Calculator

Comma-separated, for example, x+2y=5,3x+5y=14.
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Solve $\begin{cases} 5 x - 2 y = 1 \\ x + 3 y = 7 \end{cases}$ for $x$, $y$ using the Gauss-Jordan Elimination method.

## Solution

Write down the augmented matrix: $\left[\begin{array}{cc|c}5 & -2 & 1\\1 & 3 & 7\end{array}\right]$.

Perform the Gauss-Jordan elimination (for steps, see Gauss-Jordan elimination calculator): $\left[\begin{array}{cc|c}5 & -2 & 1\\0 & \frac{17}{5} & \frac{34}{5}\end{array}\right]$.

Back-substitute:

$y = \frac{\frac{34}{5}}{\frac{17}{5}} = 2$

$x = \frac{1 - \left(-2\right) \left(2\right)}{5} = 1$

$x = 1$A
$y = 2$A