If $2x + y = 23$ and $4x - y = 19$, find the value $(5y - 2x)$ and $\left(\frac{y}{x} \ -\ 2\right)$.
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)
Add eq (i) and (ii):
$2x + y + 4x - y = 23 + 19$
$6x = 42$
$x\ =\ \frac{42}{6}$
$x = 7$
Put this value of $x$ in eq (i)
$2(7) + y = 23$
$14 + y = 23$
$y = 23 - 14$
$y = 9$
After solving eq (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.
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