$$x+3y=6,5x-2y=13$$

$$x+3y=6$$

$$x=-3y+6$$

$$5\left(-3y+6\right)-2y=13$$

$$-15y+30-2y=13$$

$$-17y+30=13$$

$$-17y=-17$$

$$y=1$$

$$x=-3+6$$

$$x=3$$

$$x=3,y=1$$

$$x+3y=6,5x-2y=13$$

$$\left(\begin{matrix}1&3\\5&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\13\end{matrix}\right)$$

$$inverse(\left(\begin{matrix}1&3\\5&-2\end{matrix}\right))\left(\begin{matrix}1&3\\5&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\5&-2\end{matrix}\right))\left(\begin{matrix}6\\13\end{matrix}\right)$$

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\5&-2\end{matrix}\right))\left(\begin{matrix}6\\13\end{matrix}\right)$$

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\5&-2\end{matrix}\right))\left(\begin{matrix}6\\13\end{matrix}\right)$$

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-3\times 5}&-\frac{3}{-2-3\times 5}\\-\frac{5}{-2-3\times 5}&\frac{1}{-2-3\times 5}\end{matrix}\right)\left(\begin{matrix}6\\13\end{matrix}\right)$$

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{17}&\frac{3}{17}\\\frac{5}{17}&-\frac{1}{17}\end{matrix}\right)\left(\begin{matrix}6\\13\end{matrix}\right)$$

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{17}\times 6+\frac{3}{17}\times 13\\\frac{5}{17}\times 6-\frac{1}{17}\times 13\end{matrix}\right)$$

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\1\end{matrix}\right)$$

$$x=3,y=1$$

$$x+3y=6,5x-2y=13$$

$$5x+5\times 3y=5\times 6,5x-2y=13$$

$$5x+15y=30,5x-2y=13$$

$$5x-5x+15y+2y=30-13$$

$$15y+2y=30-13$$

$$17y=30-13$$

$$17y=17$$

$$y=1$$

$$5x-2=13$$

$$5x=15$$

$$x=3$$

$$x=3,y=1$$