Question

Решите систему, пожалуйста!
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: $xne 0,\20+xne0, xne-20. (*)$

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

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

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