Question

If x and y are non-negative integers such that $x + 9 = z, y + 1 = z$ and $x + y < z + 5$, then the maximum possible value of $2x + y$ equals

Answer 1

Given $x+9=z=y+1$ and $x+y<z+5$

$⇒ (z-9)+(z-1)<z+5$

$⇒ z<15$

Hence the maximum value of  $z=14$, max of $x=5$ and max of $y=13$

Required answer, $2x+y=2×5+13=23$

Answer 2

Given:
Equation 1. $x+9=z$
Equation 2. $y+1=z$
Equation 3. $x+y\lt z+5$
Now From equation 1 and 2:
$x=z-9$
$y=z-1$
Now put these expressions for x and y in equation 3
$(z-9)+(z-1)\lt z+5$
$2z-10\lt z+5$
$z\lt15$
Now, the maximum possible value can't be 15 or greater than 15. So, less than 15 is 14.
$z=14$
We need to maximize
$2x+y$
$2(z-9)+(z-1) \;=\;3z-19$
Now put the value of z
$3(14)-19 =23$

So, the answer is 23.