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Solve the following system of equations:
$\frac{15}{u}\ +\ \frac{2}{v}\ =\ 17$
$\frac{1}{u}\ +\ \frac{1}{v}\ =\ \frac{36}{5}$
Given:
The given system of equations is:
$\frac{15}{u}\ +\ \frac{2}{v}\ =\ 17$
$\frac{1}{u}\ +\ \frac{1}{v}\ =\ \frac{36}{5}$
To do:
We have to solve the given system of equations.
Solution:
Let $\frac{1}{u}=x$ and $\frac{1}{v}=y$
This implies,
The given system of equations can be written as,
$\frac{15}{u}\ +\ \frac{2}{v}\ =\ 17$
$15x+2y=17$-----(i)
$\frac{1}{u}\ +\ \frac{1}{v}\ =\ \frac{36}{5}$
$x+y=\frac{36}{5}$
$5x+5y=36$
$5x=36-5y$
$x=\frac{36-5y}{5}$
Substitute $x=\frac{36-5y}{5}$ in equation (i), we get,
$15(\frac{36-5y}{5})+2y=17$
$3(36-5y)+2y=17$
$108-15y+2y=17$
$-13y=17-108$
$-13y=-91$
$y=\frac{-91}{-13}$
$y=7$
This implies,
$x=\frac{36-5(7)}{5}$
$x=\frac{36-35}{5}$
$x=\frac{1}{5}$
$u=\frac{1}{x}=\frac{1}{\frac{1}{5}}=5$
$v=\frac{1}{y}=\frac{1}{7}$
Therefore, the solution of the given system of equations is $u=5$ and $v=\frac{1}{7}$.