Question

Solve 2/x-3/y=15, 8/x+5/y=77

Answer 1

$\tt{Let \: \dfrac{1}{x} \: be \: u \: and \dfrac{1}{y} \: be \: v}\\\\\\\rm{According\: to \: the \: question;}\\\\\\\sf{2u - 3v=15 \quad \dashrightarrow [Equation \: 1]}\\\\\sf{8u+5v = 77 \quad \dashrightarrow [Equation \: 2]}\\\\\\\sf{Multiplying \: Equation \: 1 \: by \: 4,}\\\\\sf{4(2u - 3v) = 4(15)}\\\\\longrightarrow \: \sf{8u - 12v = 60 \quad \dashrightarrow [Equation \: 3]}\\\\\\\sf{Subtracting \: Equation\: 3 \: from \: Equation \: 2,}$

$\quad \sf{8u + 5v = 77}\\\: \quad \sf{\underline{[-]8u - 12v = 60}}\\\\\sf{17v = 17}\\\\\implies \sf{v = \dfrac{17}{17}}\\\\\implies \boxed{\large{\bf{v = 1}}}\\\\\\\rm{Substitute \: value \: of \: v \: in \: any \: one \: of \: the \:Equations}\\\rm{to \: get \: the \: value \: of \: u}\\\\\\\sf{8u + 5v = 77}\\\\\implies \sf{8u + 5(1) = 77}\\\\\implies \sf{8u +5 = 77}\\\\\implies \sf{8u = 77-5}\\\\\implies \sf{8u = 72}\\\\\\\implies \sf{u = \dfrac{72}{8}}$

$\implies \boxed{\large{\bf{u = 9}}}\\\\\\\sf{Earlier \: we \: had \: assumed;}\\\\\star \sf{u=\dfrac{1}{x}}\\\\\star \sf{y = \dfrac{1}{v}}\\\\\implies \large{\bf{\boxed{\bf{x = \dfrac{1}{9}}}}}\\\\\\\large{\boxed{\bf{v = 1}}}$