Решите систему уравнений методом алгебраического сложения:
а) $\begin{equation*}
\begin{cases}
3x + 7y = 46, &\\
4x - 3y = 12; &
\end{cases}
\end{equation*}$
б) $\begin{equation*}
\begin{cases}
-3x + 4y = 24, &\\
5x + 3y = -40; &
\end{cases}
\end{equation*}$
в) $\begin{equation*}
\begin{cases}
5x + 3y = 20, &\\
2x - 4y = 21; &
\end{cases}
\end{equation*}$
г) $\begin{equation*}
\begin{cases}
-5x + 3y = -15, &\\
2x + 7y = 47. &
\end{cases}
\end{equation*}$
$\begin{equation*}
\begin{cases}
3x + 7y = 46 | * (-4) &\\
4x - 3y = 12 | * 3&
\end{cases}
\end{equation*}$
$\begin{equation*}
\begin{cases}
-12x - 28y = -164 &\\
12x - 9y = 36&
\end{cases}
\end{equation*}$
−12x − 28y + 12x − 9y = −184 + 36
−37y = −148
y = −148 : (−37)
y = 4
4x − 3y = 12
4x = 12 + 3y
4x = 12 + 3 * 4
4x = 12 + 12
4x = 24
x = 24 : 4
x = 6
Ответ: (6;4)
$\begin{equation*}
\begin{cases}
-3x + 4y = 24 | * 5 &\\
5x + 3y = -40 | * 3 &
\end{cases}
\end{equation*}$
$\begin{equation*}
\begin{cases}
-15x + 20y = 120 &\\
15x + 9y = -120 &
\end{cases}
\end{equation*}$
−15x + 20y + 15x + 9y = 120 − 120
29y = 0
y = 0
−3x + 4y = 24
−3x = 24 − 4y
−3x = 24 − 4 * 0
−3x = 24
x = 24 : (−3)
x = −8
Ответ: (−8;0)
$\begin{equation*}
\begin{cases}
5x + 3y = 20 | * (-2) &\\
2x - 4y = 21 | * 5&
\end{cases}
\end{equation*}$
$\begin{equation*}
\begin{cases}
-10x - 6y = -40 &\\
10x - 20y = 105 &
\end{cases}
\end{equation*}$
−10x − 6y + 10x − 20y = −40 + 105
−26y = 65
y = 65 : (−26)
y = −2,5
2x − 4y = 21
2x = 21 + 4y
2x = 21 + 4 * (−2,5)
2x = 21 − 10
2x = 11
x = 11 : 2
x = 5,5
Ответ: (5,5;−2,5)
$\begin{equation*}
\begin{cases}
-5x + 3y = -15 | * 2 &\\
2x + 7y = 47 | * 5 &
\end{cases}
\end{equation*}$
$\begin{equation*}
\begin{cases}
-10x + 6y = -30 &\\
10x + 35y = 235 &
\end{cases}
\end{equation*}$
−10x + 6y + 10x + 35y = −30 + 235
41y = 205
y = 205 : 41
y = 5
2x + 7y = 47
2x = 47 − 7y
2x = 47 − 7 * 5
2x = 47 − 35
2x = 12
x = 12 : 2
x = 6
Ответ: (6;5)
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