№ 1202 Алгебра = № 55.10 Математика
Розв’яжіть систему рівнянь:
1) $\begin{cases} 3x + 4y = 0, \\ 2x − 7y = 29; \end{cases}$
2) $\begin{cases} 8x − 5y = 41, \\ 4x + 3y = −7; \end{cases}$
3) $\begin{cases} 2a − 5b = 0, \\ −7a + 4b = 27; \end{cases}$
4) $\begin{cases} 10m − 2n = 39, \\ 9m + 4n = 38. \end{cases}$
Розв'язок:
1) $\begin{cases} 3x + 4y = 0, \\ 2x – 7y = 29; \end{cases}$ $\begin{cases} 3x = –4y, \\ 2x – 7y = 29; \end{cases}$
$\begin{cases} x = –\frac{4}{3}y, \\ 2(–\frac{4}{3}y – 7y = 29 | · –3; \end{cases}$
$\begin{cases} x = –\frac{4}{3}y, \\ 8y + 21y = –87; \end{cases}$
$\begin{cases} x = –\frac{4}{3}y, \\ 29y = –87; \end{cases}$ $\begin{cases} x = –\frac{4}{3}y, \\ y = –3; \end{cases}$
$\begin{cases} x = –\frac{4}{3}y · (–3), \\ y = –3; \end{cases}$ $\begin{cases} x = 4, \\ y = –3. \end{cases}$
2) $\begin{cases} 8x – 5y = 41, \\ 4x + 3y = –7; \end{cases}$
$\begin{cases} 5y = 8x – 41, \\ 4x + 3y = –7; \end{cases}$
$\begin{cases} y = \frac{8x – 41}{5}, \\ 4x + 3\frac{8x – 41}{5} = –7 | · 5; \end{cases}$
$\begin{cases} y = \frac{8x – 41}{5}, \\ 20x + 3(8x –41) = –35; \end{cases}$
$\begin{cases} y = \frac{8x – 41}{5}, \\ 20x + 24x – 123 = –35; \end{cases}$
$\begin{cases} y = \frac{8x – 41}{5}, \\ 44x = 88; \end{cases}$ $\begin{cases} y = \frac{8x – 41}{5}, \\ x = 2; \end{cases}$
$\begin{cases} y = \frac{8 · 2 – 41}{5}, \\ x = 2; \end{cases}$ $\begin{cases} y = –5, \\ x = 2.\end{cases}$
3) $\begin{cases} 2a – 5b = 0, \\ –7a + 4b = 27; \end{cases}$
$\begin{cases} 2a = 5b, \\ –7a + 4b = 27; \end{cases}$
$\begin{cases} a = \frac{5}{2}b, \\ –7\frac{5}{2}b + 4b = 27 | · 2; \end{cases}$
$\begin{cases} a = \frac{5}{2}b, \\ –35b + 8b = 54; \end{cases}$
$\begin{cases} a = \frac{5}{2}b, \\ –27b = 54; \end{cases}$ $\begin{cases} a = \frac{5}{2}b, \\ b = –2; \end{cases}$
$\begin{cases} a = \frac{5}{2}b, \\ b = –2; \end{cases}$ $\begin{cases} a = –5, \\ b = –2. \end{cases}$
4) $\begin{cases} 10m – 2n = 9, \\ 9m + 4n = 38; \end{cases}$
$\begin{cases} 2n = 10m – 39, \\ 9m + 4n = 38; \end{cases}$
$\begin{cases} n = \frac{10m – 39}{2}, \\ 9m + 4\frac{10m – 39}{2} = 38; \end{cases}$
$\begin{cases} n = \frac{10m – 39}{2}, \\ 9m + 20m – 78 = 38; \end{cases}$
$\begin{cases} n = \frac{10m – 39}{2}, \\ 29m = 116; \end{cases}$ $\begin{cases} n = \frac{10m – 39}{2}, \\ m = 4; \end{cases}$
$\begin{cases} n = \frac{10 · 4 – 39}{2}, \\ m = 4; \end{cases}$ $\begin{cases} n = \frac{1}{2}, \\ m = 4.\end{cases}$
Відповідь:
1) (4; −3);
2) (2; −5);
3) (−5; −2);
4) ($\frac{1}{2}$; 4).