Given that,
$ \frac{x+y+z}{3} = 80 $
$ \Rightarrow x+y+z = 240 \quad \longrightarrow (1) $
And, $\frac{x+y+z+u+v}{5} = 75 $
$ \Rightarrow x+y+z+u+v = 375 \quad \longrightarrow {2} $
Also, $ u = (x+y)/2 \; , \; v = (y+z)/2 $
Put the value of $u$ and $v$ in the equation $(2),$ we get
$ x+y+z+\frac{x+y}{2}+\frac{y+z}{2} = 375 $
$ \Rightarrow \frac{2(x+y+z) + x+y+z+y}{2} = 375 $
$ \Rightarrow 2(240) + 240 + y = 750 $
$ \Rightarrow 720 + y = 750 $
$ \Rightarrow \boxed{y = 30} $
Put the value of $y$ in equation $(1),$ we get
$ x+y+z = 240 $
$ \Rightarrow x+30+z = 240 $
$ \Rightarrow x+z = 240 – 30 $
$ \Rightarrow x+z = 210 \quad \longrightarrow (3) $
Since, $ x \geqslant z, \; x $ takes the minimum possible value, when $ x = z.$
From equation $(3),$
$ x+z = 210 $
$ \Rightarrow x+x = 210 $
$ \Rightarrow 2x = 210 $
$ \Rightarrow \boxed{x = 105} $
$\therefore$ The minimum possible value of $x$ is $105.$
Correct Answer $:105$
$\textbf{PS:}$ The arithmetic mean is the sum of all the numbers in a data set divided by the quantity of numbers in that set.
The arithmetic mean $\overline{x}$ of a collection of $n$ numbers $(\text{from}\; a_1$ through $a_n)$ is given by the formula:
$$\overline{x}=\displaystyle \frac{1}{n}\sum_{i=1}^n a_i = \frac{a_1+a_2+a_3+\dots + a_n}{n}.\ _\square$$
Reference: https://brilliant.org/wiki/arithmetic-mean/