$\estwo{2 \subscript{x}{1} + 3 \subscript{x}{2} = 7}{4 \subscript{x}{1} - 4 \subscript{x}{2} = -6}$

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2x_{1}+3x_{2}=7,4x_{1}-4x_{2}=-6

2x_{1}+3x_{2}=7

2x_{1}=-3x_{2}+7

x_{1}=\frac{1}{2}\left(-3x_{2}+7\right)

x_{1}=-\frac{3}{2}x_{2}+\frac{7}{2}

4\left(-\frac{3}{2}x_{2}+\frac{7}{2}\right)-4x_{2}=-6

-6x_{2}+14-4x_{2}=-6

-10x_{2}+14=-6

-10x_{2}=-20

x_{2}=2

x_{1}=-\frac{3}{2}\times 2+\frac{7}{2}

x_{1}=-3+\frac{7}{2}

x_{1}=\frac{1}{2}

x_{1}=\frac{1}{2},x_{2}=2

2x_{1}+3x_{2}=7,4x_{1}-4x_{2}=-6

\left(\begin{matrix}2&3\\4&-4\end{matrix}\right)\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}7\\-6\end{matrix}\right)

inverse(\left(\begin{matrix}2&3\\4&-4\end{matrix}\right))\left(\begin{matrix}2&3\\4&-4\end{matrix}\right)\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\4&-4\end{matrix}\right))\left(\begin{matrix}7\\-6\end{matrix}\right)

\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\4&-4\end{matrix}\right))\left(\begin{matrix}7\\-6\end{matrix}\right)

\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\4&-4\end{matrix}\right))\left(\begin{matrix}7\\-6\end{matrix}\right)

\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{-4}{2\left(-4\right)-3\times 4}&-\frac{3}{2\left(-4\right)-3\times 4}\\-\frac{4}{2\left(-4\right)-3\times 4}&\frac{2}{2\left(-4\right)-3\times 4}\end{matrix}\right)\left(\begin{matrix}7\\-6\end{matrix}\right)

\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{3}{20}\\\frac{1}{5}&-\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}7\\-6\end{matrix}\right)

\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\times 7+\frac{3}{20}\left(-6\right)\\\frac{1}{5}\times 7-\frac{1}{10}\left(-6\right)\end{matrix}\right)

\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\\2\end{matrix}\right)

x_{1}=\frac{1}{2},x_{2}=2

2x_{1}+3x_{2}=7,4x_{1}-4x_{2}=-6

4\times 2x_{1}+4\times 3x_{2}=4\times 7,2\times 4x_{1}+2\left(-4\right)x_{2}=2\left(-6\right)

8x_{1}+12x_{2}=28,8x_{1}-8x_{2}=-12

8x_{1}-8x_{1}+12x_{2}+8x_{2}=28+12

12x_{2}+8x_{2}=28+12

20x_{2}=28+12

20x_{2}=40

x_{2}=2

4x_{1}-4\times 2=-6

4x_{1}-8=-6

4x_{1}=2

x_{1}=\frac{1}{2}

x_{1}=\frac{1}{2},x_{2}=2