1/2x+1/3y=2,1/3x+1/2y=13/6

{tex}{\frac{1}{{2x}} + \frac{1}{{3y}} = 2}{/tex} ... (1)
{tex}\frac{1}{{3x}} + \frac{1}{{2y}} = \frac{{13}}{6}{/tex} ...(2)
Let {tex}\frac{1}{x}{/tex}= p and {tex}\frac{1}{y}{/tex}= q
Putting this in equation (1) and (2), we get
{tex}\frac{p}{2} + \frac{q}{3} = 2{/tex} and {tex}\;\frac{p}{3} + \frac{q}{2} = \frac{{13}}{6}{/tex}
{tex}\Rightarrow{/tex} 3p + 2q = 12 and 6(2p +3q) = 13 (6)
{tex}\Rightarrow{/tex} 3p + 2q = 12 and 2p + 3q = 13
{tex}\Rightarrow{/tex} 3p + 2q - 12 = 0 .................. (3) and
2p + 3q - 13 = 0 ................. (4)

{tex}\frac{p}{{2( - 13) - 3( - 12)}} = \frac{q}{{( - 12)2 - ( - 13)3}}{/tex}{tex} = \frac{1}{{3 \times 3 - 2 \times 2}}{/tex}
{tex}\Rightarrow \frac{p}{{ - 26 + 36}} = \frac{q}{{ - 24 + 39}} = \frac{1}{{9 - 4}}{/tex}
{tex} \Rightarrow \frac{p}{{10}} = \frac{q}{{15}} = \frac{1}{5} \Rightarrow \frac{p}{{10}} = \frac{1}{5}\,{\text{and}}\,\frac{q}{{15}} = \frac{1}{5}{/tex}
{tex}\Rightarrow{/tex} p = 2 and q = 3
But {tex}\frac{1}{x}{/tex} = p and {tex}\frac{1}{y}{/tex} = q
Putting value of p and q in this we get,
x = {tex}\frac{1}{2}{/tex}and y = {tex}\frac{1}{2}{/tex}