Question

Решить системы уравнений:
1) x - 2y = 4
3x + 5y = 10
2) 7x + 3y = 2
2x - 15y = 1
3) x - 7y = 2
3y - 2x = 8

Answer 1

$1) \; \left\{\begin{array}{ccc}x-2y=4\\3x+5y=10\end{array}\right\;\Longrightarrow\;\;\;\left\{\begin{array}{ccc}x=4+2y\\3(4+2y)+5y=10\end{array}\right\\\\3(4+2y)+5y=10\\12+6y+5y=10\\12+11y=10\\11y=-2\\y=-2:11\\y=-\dfrac2{11}\\\\\\x=4+2y\\x=4+2\cdot(-\dfrac2{11})\\x=4-\dfrac{4}{11}\\\\x=3\dfrac{7}{11}\\\\x=\dfrac{40}{11}\\\\\\Om\beta em: \; (\dfrac{40}{11};\;-\dfrac{2}{11})$

$2)\;\left\{\begin{array}{ccc}7x+3y=2\\2x-15y=1\end{array}\right\Rightarrow\left\{\begin{array}{ccc}3y=2-7x\\2x-15y=1\end{array}\right\Rightarrow\left\{\begin{array}{ccc}y=\dfrac23-\dfrac73x\\\\2x-15\cdot(\dfrac23-\dfrac73x)=1\end{array}\right$

$2x-15\cdot(\dfrac23-\dfrac73x)=1\\\\2x-\dfrac{30}3+\dfrac{105}{3}x=1\\\\2x-10+35x=1\\37x-10=1\\37x=11\\\\x=\dfrac{11}{37}\\\\\\y=\dfrac23-\dfrac73\cdot\dfrac{11}{37}\\\\y=\dfrac23-\dfrac{77}{111}\\\\y=\dfrac{74}{111}-\dfrac{77}{111}\\\\y=-\dfrac{3}{111}\\\\y=-\dfrac{1}{37}\\\\Om\beta em:\;(\dfrac{11}{37};\;-\dfrac{1}{37})$

$\left\{\begin{array}{ccc}x-7y=2\\3y-2x=8\end{array}\right\;\Longrightarrow\;\;\;\left\{\begin{array}{ccc}x=2+7y\\3y-2(2+7y)=8\end{array}\right\\\\3y-2(2+7y)=8\\3y-4-14y=8\\-11y-4=8\\11y+4=-8\\11y=-12\\y=-\dfrac{12}{11}\\\\\\x=2+7(-\dfrac{12}{11})\\\\x=2-\dfrac{84}{11}\\\\x=\dfrac{22}{11}-\dfrac{84}{11}\\\\x=-\dfrac{62}{11}\\\\\\Om\beta em:\;(-\dfrac{62}{11};\;-\dfrac{12}{11})$