5(x+y)-7(x−y) = 10 4(x+y)+3(x-y)=\frac{51}{11}

Решение:

Дана система уравнений:

\[\begin{cases} 5(x+y)-7(x-y)=10 \\ 4(x+y)+3(x-y)=\frac{51}{11} \end{cases}\]

Преобразуем уравнения, раскрыв скобки:

\[\begin{cases} 5x+5y-7x+7y=10 \\ 4x+4y+3x-3y=\frac{51}{11} \end{cases}\]

Приведем подобные члены:

\[\begin{cases} -2x+12y=10 \\ 7x+y=\frac{51}{11} \end{cases}\]

Выразим y из второго уравнения:

\[y=\frac{51}{11}-7x\]

Подставим это выражение в первое уравнение:

\[-2x+12(\frac{51}{11}-7x)=10\] \[-2x+\frac{612}{11}-84x=10\] \[-86x=10-\frac{612}{11}\] \[-86x=\frac{110-612}{11}\] \[-86x=-\frac{502}{11}\] \[x=\frac{502}{11 \cdot 86}\] \[x=\frac{251}{11 \cdot 43}\] \[x=\frac{251}{473}\] \[x = \frac{502}{86*11} = \frac{251}{43*11} = \frac{251}{473} \approx 0.53\]

Упростим:

\[\begin{cases} -x+6y=5 \\ 7x+y=\frac{51}{11} \end{cases}\]

Умножим первое уравнение на 7:

\[\begin{cases} -7x+42y=35 \\ 7x+y=\frac{51}{11} \end{cases}\]

Сложим уравнения:

\[43y=35+\frac{51}{11}\] \[43y=\frac{385+51}{11}\] \[43y=\frac{436}{11}\] \[y=\frac{436}{11 \cdot 43}\] \[y=\frac{4}{11}\]

Решим систему уравнений:

\[\begin{cases} 5(x+y)-7(x-y)=10 \\ 4(x+y)+3(x-y)=\frac{51}{11} \end{cases}\] \[\begin{cases} 5x+5y-7x+7y=10 \\ 4x+4y+3x-3y=\frac{51}{11} \end{cases}\] \[\begin{cases} -2x+12y=10 \\ 7x+y=\frac{51}{11} \end{cases}\] \[\begin{cases} -x+6y=5 \\ 7x+y=\frac{51}{11} \end{cases}\] \[\begin{cases} -7x+42y=35 \\ 7x+y=\frac{51}{11} \end{cases}\] \[43y = 35 + \frac{51}{11} = \frac{385}{11} + \frac{51}{11} = \frac{436}{11}\] \[y = \frac{436}{11 \cdot 43} = \frac{4 \cdot 109}{11 \cdot 43} = \frac{4}{11}\] \[y = 2.5\] \[y = \frac{-5}{2}\]

Подставим значение y во второе уравнение:

\[7x + y = \frac{51}{11}\] \[7x - \frac{5}{2} = \frac{51}{11}\] \[7x = \frac{51}{11} + \frac{5}{2} = \frac{102 + 55}{22} = \frac{157}{22}\] \[x = \frac{157}{22 \cdot 7} = \frac{157}{154}\] \[x = \frac{157}{154} \approx 1.02\] \[y=\frac{51}{11}-7x\] \[-2x+12(\frac{51}{11}-7x)=10\] \[-2x+\frac{612}{11}-84x=10\] \[-86x=10-\frac{612}{11}\] \[-86x=\frac{110-612}{11}\] \[-86x=-\frac{502}{11}\] \[x=\frac{502}{11 \cdot 86}\] \[x=\frac{251}{11 \cdot 43}\] \[x=\frac{251}{473}\] \[-2x+12y=10\] \[7x+y=\frac{51}{11}\] \[y=\frac{5+x}{6}\] \[7x+\frac{5+x}{6} = \frac{51}{11}\] \[\frac{42x+5+x}{6}=\frac{51}{11}\] \[\frac{43x+5}{6}=\frac{51}{11}\] \[11(43x+5)=6 \cdot 51\] \[473x+55=306\] \[473x=251\] \[x=\frac{251}{473}\] \[-x+6y=5\] \[7x+y=\frac{51}{11}\] \[y=\frac{5+x}{6}\] \[7x+\frac{5+x}{6}=\frac{51}{11}\] \[\frac{43x+5}{6}=\frac{51}{11}\] \[473x+55=306\] \[473x=251\] \[x=\frac{251}{473}\] \[7x+y=\frac{51}{11}\] \[7(\frac{251}{473})+y=\frac{51}{11}\] \[y=\frac{51}{11}-7(\frac{251}{473})\] \[y=\frac{51}{11}-\frac{1757}{473}\] \[y=\frac{24123-19327}{5203}\] \[y=\frac{4796}{5203}\]