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

The correct answer is 23.

We are given three conditions involving non-negative integers x and y, and a variable z:

   ①  $x+9=z$
   ②  $y+1=z$
   ③  $x+y<z+5$

Step 1 — Express y in terms of x.

Since both ① and ② equal z, we can set them equal to each other:

$x+9=y+1$

Solving for y:

$y=x+8$

Step 2 — Apply the inequality constraint ③.

Substitute $z=x+9$ into ③:

$x+y<(x+9)+5$

$x+y<x+14$

Subtract x from both sides:

$y<14$

Since y must be a non-negative integer, the maximum value of y is 13.

Step 3 — Find the corresponding value of x.

Using $y=x+8$ and setting $y=13$:

$13=x+8 ⟹ x=5$

Step 4 — Verify all conditions are satisfied.

With $x=5$, $y=13$, and $z=14$:

   ①  $x+9=5+9=14=z$  ✓
   ②  $y+1=13+1=14=z$  ✓
   ③  $x+y=18<z+5=19$  ✓

Step 5 — Compute the maximum value of 2x + y.

$2x+y=2\left(5\right)+13=10+13=23$

Note that to maximise $2x+y$, we can write it in terms of x alone. Since $y=x+8$:

$2x+y=2x+(x+8)=3x+8$

This is increasing in x, so we want x as large as possible. The constraint $y<14$ forces $x\le 5$, confirming $x=5$ is optimal.

Therefore, the maximum possible value of $2x+y$ is 23.