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

Given that, $x$ and $y$ are non-negative integers. That means  $x \geq 0, y \geq 0.$

And,

• $x + 9 = z \Rightarrow x = z – 9 \; \longrightarrow (1)$
• $y + 1 = z \Rightarrow y = z – 1 \; \longrightarrow (2)$
• $x + y < z + 5 \; \longrightarrow (3)$

Put the value of $x,$ and $y$ in the equation $(3),$ we get.

$x + y < z + 5$

$\Rightarrow (z – 9) + (z – 1) < z + 5$

$\Rightarrow z – 10 < 5$

$\Rightarrow \boxed{z < 15}$

Maximum value of $z$ can be $14.$

So,

• $x_{\textsf{max}} = 14 – 9 = 5$
• $y_{\textsf{max}} = 14 – 1 = 13$

Thus, the value of $2x+y = 2(5) + 13 = 23.$

$\therefore$ The maximum possible value of $2x+y$ is $23.$

Correct Answer$: 23$

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