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$\left\{ \begin{array} { l } { 2 x + y = 6 } \\ { 4 x - y = 7 } \end{array} \right.$

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2x+y=6,4x-y=7

2x+y=6

2x=-y+6

x=\frac{1}{2}\left(-y+6\right)

x=-\frac{1}{2}y+3

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

-2y+12-y=7

-3y+12=7

-3y=-5

y=\frac{5}{3}

x=-\frac{1}{2}\times \left(\frac{5}{3}\right)+3

x=-\frac{5}{6}+3
-\frac{1}{2} 乘以 \frac{5}{3} 的计算方法是，将两数分子与分子相乘得到分子，分母与分母相乘得到分母。如果可能，将所得分数化为最简分数。
x=\frac{13}{6}

x=\frac{13}{6},y=\frac{5}{3}

2x+y=6,4x-y=7

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

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

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

\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{-1}{2\left(-1\right)-4}&-\frac{1}{2\left(-1\right)-4}\\-\frac{4}{2\left(-1\right)-4}&\frac{2}{2\left(-1\right)-4}\end{matrix}\right)\left(\begin{matrix}6\\7\end{matrix}\right)

\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&\frac{1}{6}\\\frac{2}{3}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}6\\7\end{matrix}\right)

\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\times 6+\frac{1}{6}\times 7\\\frac{2}{3}\times 6-\frac{1}{3}\times 7\end{matrix}\right)

\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{6}\\\frac{5}{3}\end{matrix}\right)

x=\frac{13}{6},y=\frac{5}{3}

2x+y=6,4x-y=7

4\times 2x+4y=4\times 6,2\times 4x+2\left(-1\right)y=2\times 7

8x+4y=24,8x-2y=14

8x-8x+4y+2y=24-14

4y+2y=24-14

6y=24-14

6y=10

y=\frac{5}{3}

4x-\frac{5}{3}=7

4x=\frac{26}{3}

x=\frac{13}{6}

x=\frac{13}{6},y=\frac{5}{3}