# How do I use Cramer's rule to solve the system of equations 3x-5y=-31 and 2x+y=1?

Oct 20, 2014

Convert the system of linear equations

$\left\{\begin{matrix}3 x - 5 y = - 31 \\ 2 x + y = 1\end{matrix}\right.$

into the matrix equation

$\left[\begin{matrix}3 \text{ "-5 \\ 2" "" } 1\end{matrix}\right] \left[\begin{matrix}x \\ y\end{matrix}\right] = \left[\begin{matrix}- 31 \\ 1\end{matrix}\right]$.

Let u s now find the necessary determinants

The determinant of the coefficient matrix is

$| \left(3 \text{ "-5),(2" "" } 1\right) | = 3 \cdot 1 - \left(- 5\right) \cdot 2 = 13$

By replacing the first column by the right-hand side,

$| \left(- 31 \text{ "-5),(" "1" "" } 1\right) | = \left(- 31\right) \cdot 1 - \left(- 5\right) \cdot 1 = - 26$.

By replacing the second column by the right-hand side,

$| \left(3 \text{ "-31),(2" "" } 1\right) | = 3 \cdot 1 - \left(- 31\right) \cdot 2 = 65$

{(x={-26}/{13}=-2), (y={65}/{13}=5):}

I hope that this was helpful.