Solve the following system of equations:

$\frac{x}{7}\ +\ \frac{y}{3}\ =\ 5$
$\frac{x}{2}\ –\ \frac{y}{9}\ =\ 6$

Given:

The given system of equations is:

$\frac{x}{7}\ +\ \frac{y}{3}\ =\ 5$

$\frac{x}{2}\ –\ \frac{y}{9}\ =\ 6$

To do:

We have to solve the given system of equations.

Solution:

The given system of equations can be written as,

$\frac{x}{7}+\frac{y}{3}=5$

$\Rightarrow \frac{3(x)+7(y)}{21}=5$

$\Rightarrow 3x+7y=5(21)$   (On cross  multiplication)

$\Rightarrow 3x+7y=105$---(i)

$\frac{x}{2}-\frac{y}{9}=6$

$\Rightarrow \frac{9(x)-2(y)}{18}=6$

$\Rightarrow 9x-2y=6(18)$   (On cross multiplication)

$\Rightarrow 9x=2y+108$

$\Rightarrow x=\frac{2y+108}{9}$----(ii)

Substitute $x=\frac{2y+108}{9}$ in equation (i), we get,

$3(\frac{2y+108}{9})+7y=105$

$\frac{2y+108}{3}+7y=105$ 

Multiplying by $3$ on both sides, we get,

$3(\frac{2y+108}{3})+3(7y)=3(105)$

$2y+108+21y=315$

$23y=315-108$

$23y=207$

$y=\frac{207}{23}$

$y=9$

Substituting the value of $y=9$ in equation (ii), we get,

$x=\frac{2(9)+108}{9}$

$x=\frac{18+108}{9}$

$x=\frac{126}{9}$

$x=14$

Therefore, the solution of the given system of equations is $x=14$ and $y=9$.