Question

Solve for x and y: $\frac{2}{x-1}+\frac{3}{y-2}=1$ and $\frac{5}{x-1}+\frac{6}{y-2}=7$

Answer 1

$let\left(\frac{1}{x-1}\right)=uand\left(\frac{1}{y-2}\right)=vthen2u+3v=1\to \left(1.\right)and5u+6v=7\to \left(2.\right)multiplye{q}^n\left(1\right)by2,weget4u+6v=2\to \left(1.\right)5u+6v=7\to \left(2.\right)subtract\left(1.\right)-\left(2.\right),weget-u+0=-5\therefore u=5thenv=\left(\frac{2-4u}{6}\right)=\left(\frac{-18}{6}\right)=-3\therefore \left(\frac{1}{x-1}\right)=5orx-1=\left(\frac{1}{5}\right)\therefore x=\left(\frac{6}{5}\right)and\left(\frac{1}{y-2}\right)=-3ory-2=\left(\frac{-1}{3}\right)\therefore y=\left(\frac{5}{3}\right)$