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 $2 x+y$ ?

We're given three key pieces of information:

$x + 9 = z$ ... (equation 1)

$y + 1 = z$ ... (equation 2)

$x + y < z + 5$ ... (inequality)

Key insight: Since both $x + 9$ and $y + 1$ equal $z$, we can use this to connect $x$ and $y$.

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From equations (1) and (2):

$x = z - 9$

$y = z - 1$

Adding these equations together:

$(x + 9) + (y + 1) = z + z$

$x + y + 10 = 2z$

Therefore: $x + y = 2z - 10$

This expresses $x + y$ in terms of $z$ only, which will help us use the inequality.

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We know that $x + y < z + 5$

Substituting our expression for $x + y$:

$2z - 10 < z + 5$

$2z - z < 5 + 10$

$z < 15$

Since $z$ must be an integer and $z < 15$, the maximum possible value is $z = 14$.

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Once we know $z = 14$:

From equation (1): $x = z - 9 = 14 - 9 = 5$

From equation (2): $y = z - 1 = 14 - 1 = 13$

Let's check our constraint $x + y < z + 5$:

$5 + 13 = 18$ and $14 + 5 = 19$

Indeed, $18 < 19$

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Maximum value of $2x + y = 2(5) + 13 = 10 + 13 = 23$

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The key insight is that to maximize $2x + y$, we need to maximize both $x$ and $y$. Since both $x$ and $y$ increase as $z$ increases, we want the largest possible $z$ that satisfies our constraints.

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