Question

Решите систему, ! 90/x-90/y = 5/12 y-x = 20

Answer 1

$\left \{ {{\frac{90}{x}-\frac{90}{y}=\frac{5}{12}} \atop {y-x=20}} \right. \\\\1)\ y-x=20,\ y=20+x;\\\\2)\ \frac{90}{x}-\frac{90}{20+x}=\frac{5}{12},$

Область определения уравнения: $x\ne 0,\\20+x\ne0,\ x\ne-20.\ (*)$

$\frac{90(20+x)-90x}{x(20+x)}-\frac{5}{12}=0,\\\\\frac{12(1800+90x-90x)-5x(20+x)}{12x(20+x)}=0,\\\\\frac{21600-100x-5x^2}{12x(20+x)}=0,\\\\21600-100x-5x^2=0\ |:(-5),\\\\x^2+20x-4320=0,\\\\D=20^2-4\bullet(-4320)=400+17280=17680\\\\x_1=\frac{-20+\sqrt{17860}}{2}=\frac{-20+4\sqrt{1105}}{2}=2\sqrt{1105}-10,\\ x_2=\frac{-20-\sqrt{17860}}{2}=-2\sqrt{1105}-10.$

$y_1=20+2\sqrt{1105}-10=10+2\sqrt{1105}=2(5+\sqrt{1105}),\\y_2=2(5-\sqrt{1105}).$

ответ: $(2\sqrt{1105}-10;\ 2\sqrt{1105}+10),\ (-2\sqrt{1105}-10;\ -2\sqrt{1105}+10).$