If $ 2 x+y=23 $ and $ 4 x-y=19 $, find the values of $ 5 y-2 x $ and $ \frac{y}{x}-2 $.

Given:

$2x + y = 23$ and $4x - y = 19$

To find:

$(5y - 2x)$ and $\left(\frac{y}{x} \ -\ 2\right)$

Solution:

$2x + y = 23$   ...(i)

$4x - y = 19$    ...(ii)

Adding (i) and (ii), we get,

$2x + y + 4x - y = 23 + 19$

$6x = 42$

$x\ =\ \frac{42}{6}$

$x = 7$

Putting this value of $x$ in (i), we get,

$2(7) + y = 23$ 

$14 + y = 23$ 

$y = 23 - 14$ 

$y = 9$ 

After solving (i) and (ii) we get the values of $x$ and $y$. Using them to calculate the values of $(5y - 2x)$ and $\left(\frac{y}{x} \ -\ 2\right)$.

a) $5y - 2x$

$= 5(9) - 2(7)$

$= 45 - 14$

$= \mathbf{31}$

b) $\left(\frac{y}{x} \ -\ 2\right)$

$= \left(\frac{9}{7} \ -\ 2\right)$

$= \left(\frac{9\ -\ 14}{7}\right)$

$= \mathbf{-\frac{5}{7}}$

So, values of $(5y - 2x)$ and $\left(\frac{y}{x} \ -\ 2\right)$ are  $\mathbf{31}$  and  $\mathbf{-\frac{5}{7}}$  respectively.