Solve the following system of equations:
$\frac{2}{x}\ +\ \frac{5}{y}\ =\ 1$
$\frac{60}{x}\ +\ \frac{40}{y}\ =\ 19$

Given:

The given system of equations is:

$\frac{2}{x}\ +\ \frac{5}{y}\ =\ 1$

$\frac{60}{x}\ +\ \frac{40}{y}\ =\ 19$

To do:

We have to solve the given system of equations.

Solution:

Let $\frac{1}{x}=u$ and $\frac{1}{y}=v$

This implies,

The given system of equations can be written as,

$\frac{2}{x}\ +\ \frac{5}{y}\ =\ 1$

$2u+5v=1$-----(i)

$\frac{60}{x}\ +\ \frac{40}{y}\ =\ 19$

$60u+40v=19$

$60u=19-40v$

$u=\frac{19-40v}{60}$

Substitute $u=\frac{19-40v}{60}$ in equation (i), we get,

$2(\frac{19-40v}{60})+5v=1$

$\frac{19-40v}{30}+5v=1$

Multiplying both sides by $30$, we get,

$30(\frac{19-40v}{30})+30(5v)=30(1)$ 

$19-40v+150v=30$ 

$110v=30-19$

$110v=11$ 

$v=\frac{11}{110}$

$v=\frac{1}{10}$

This implies,

$u=\frac{19-40(\frac{1}{10})}{60}$

$u=\frac{19-4}{60}$

$u=\frac{15}{60}$

$u=\frac{1}{4}$

$x=\frac{1}{u}=\frac{1}{\frac{1}{4}}=4$

$y=\frac{1}{v}=\frac{1}{\frac{1}{10}}=10$ 

Therefore, the solution of the given system of equations is $x=4$ and $y=10$.