Решите систему уравнений:
а) $\begin{equation*} \begin{cases} \frac{5x - 3 + 9y}{3} = \frac{2x + 3y - 2}{2}, &\\ \frac{x - 3y}{2} = \frac{2x - 3y}{3}; & \end{cases} \end{equation*}$
б) $\begin{equation*} \begin{cases} \frac{2x - y}{6} + \frac{2x + y}{9} = 3, &\\ \frac{x + y}{3} - \frac{x - y}{4} = 4; & \end{cases} \end{equation*}$
в) $\begin{equation*} \begin{cases} \frac{x + 3 - 5y}{2} = \frac{3x - 4y + 3}{3}, &\\ \frac{6 + 3x - y}{3} = \frac{12x - y}{4}; & \end{cases} \end{equation*}$
г) $\begin{equation*} \begin{cases} \frac{x + y}{8} + \frac{x - y}{6} = 5, &\\ \frac{x + y}{4} + \frac{x - y}{5} = 10. & \end{cases} \end{equation*}$

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Решение

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ГДЗ учебник по алгебре 7 класс Мордкович. §12. Метод постановки. Номер №12.22.

Решение а

$\begin{equation*} \begin{cases} \frac{5x - 3 + 9y}{3} = \frac{2x + 3y - 2}{2} &\\ \frac{x - 3y}{2} = \frac{2x - 3y}{3} & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 2(5x - 3 + 9y) = 3(2x + 3y - 2) &\\ 3(x - 3y) = 2(2x - 3y) & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 10x - 6 + 18y = 6x + 9y - 6 &\\ 3x - 9y = 4x - 6y & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 10x - 18y - 6x - 9y = 6 - 6 &\\ 3x - 9y - 4x + 6y = 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 4x - 27y = 0 &\\ -x - 3y = 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 4x - 27y = 0 &\\ x = -3y & \end{cases} \end{equation*}$
4x − 27y = 0
4 * (−3y) − 27y = 0
−12y − 27y = 0
−29y = 0
y = 0
x = −3y = −3 * 0 = 0
Ответ: (0;0)

Решение б

$\begin{equation*} \begin{cases} \frac{2x - y}{6} + \frac{2x + y}{9} = 3 | *18 &\\ \frac{x + y}{3} - \frac{x - y}{4} = 4 | *12 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 3(2x - y) + 2(2x + y) = 54 &\\ 4(x + y) - 3(x - y) = 48 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 6x - 3y + 4x + 2y = 54 &\\ 4x + 4y - 3x + 3y = 48 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 10x - y = 54 &\\ x + 7y = 48 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} y = 10x - 54 &\\ x + 7y = 48 & \end{cases} \end{equation*}$
x + 7y = 48
x + 7(10x − 54) = 48
x + 70x − 378 = 48
71x = 48 + 378
x = 426 : 71
x = 6
y = 10x − 54 = 10 * 6 − 54 = 60 − 54 = 6
Ответ: (6;6)

Решение в

$\begin{equation*} \begin{cases} \frac{x + 3 - 5y}{2} = \frac{3x - 4y + 3}{3} &\\ \frac{6 + 3x - y}{3} = \frac{12x - y}{4} & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 3(x + 3 - 5y) = 2(3x - 4y + 3) &\\ 4(6 + 3x - y) = 3(12x - y) & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 3x + 9 - 15y = 6x - 8y + 6 &\\ 24 + 12x - 4y = 36x - 3y & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 3x - 6x - 15y + 8y = 6 - 9 &\\ 12x - 4y - 36x + 3y = -24 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} -3x - 7y = -3 &\\ -24x - y = -24 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} -3x - 7y = -3 &\\ y = -24x + 24 & \end{cases} \end{equation*}$
−3x − 7y = −3
−3x − 7(−24x + 24) = −3
−3x + 168x − 168 = −3
165x = −3 + 168
165x = 165
x = 1
y = −24x + 24 = −24 * 1 + 24 = 0
Ответ: (1;0)

Решение г

$\begin{equation*} \begin{cases} \frac{x + y}{8} + \frac{x - y}{6} = 5 | *24 &\\ \frac{x + y}{4} + \frac{x - y}{5} = 10 | *20 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 3(x + y) + 4(x - y) = 120 &\\ 5(x + y) + 4(x - y) = 200 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 3x + 3y + 4x - 4y = 120 &\\ 5x + 5y + 4x - 4y = 200 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 7x - y = 120 &\\ 9x + y = 200 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 7x - y = 120 &\\ y = 200 - 9x & \end{cases} \end{equation*}$
7x − y = 120
7x − (200 − 9x) = 120
7x − 200 + 9x = 120
16x = 120 + 200
16x = 320
x = 320 : 16
x = 20
y = 200 − 9x = 200 − 9 * 20 = 200 − 180 = 20
Ответ: (20;20)