Solve 2/x+3/y=13 5/x+3/y=2

Putting {tex}\frac 1x{/tex}= u and {tex}\frac 1y{/tex} = v, the given equations become
{tex}2u+ 3v = 13{/tex} .... (i)
{tex}5u - 4v = -2{/tex} .......(ii)
{tex}\text{Multiplying (i) by 4 and (ii) by 3 and adding the results, we get}{/tex}
{tex}8u + 15u = 52 - 6{/tex}
{tex}\Rightarrow{/tex} {tex}23u = 46{/tex}
{tex}\Rightarrow \quad u = \frac { 46 } { 23 } = 2{/tex}
{tex}\text{Putting u = 2 in (i), we get}{/tex}
(2 {tex}\times {/tex} 2) + 3v = 13 {tex}\Rightarrow{/tex} 3v = 13 - 4 = 9 {tex}\Rightarrow{/tex} v = 3.
Now,u = 2{tex}\Rightarrow \frac { 1 } { x } = 2 \Rightarrow 2 x = 1 \Rightarrow x = \frac { 1 } { 2 }{/tex}
And, v = 3 {tex}\Rightarrow \frac { 1 } { y } = 3 \Rightarrow 3 y = 1 \Rightarrow y = \frac { 1 } { 3 }{/tex}
Hence, {tex}x = \frac { 1 } { 2 } \text { and } y = \frac { 1 } { 3 }{/tex}