Given the ratio $x : y : z = 8 : 10 : 17$, we can write:

$x = 8k, \quad y = 10k, \quad z = 17k$

where $k$ is a constant.

Since $z - y = 84$, we substitute the values for $z$ and $y$:

$17k - 10k = 84$

This simplifies to:

$7k = 84$

Dividing both sides by 7:

$k = 12$

Now, we substitute $k = 12$ back into the expressions for $x$ and $z$:

$x = 8 \times 12 = 96, \quad z = 17 \times 12 = 204$

Therefore, $x + z = 96 + 204 = 300$.

Answer:

$x + z = 300$

Would you like further details on the solution steps?

Here are related questions for practice:

1. If $x : y : z = 5 : 7 : 9$ and $z - x = 40$, find $x + y$.
2. Given $a : b : c = 3 : 4 : 5$ and $c - b = 20$, calculate $a + c$.
3. For $p : q : r = 6 : 11 : 15$ and $r - q = 44$, determine $p + r$.
4. If $m : n : o = 2 : 3 : 7$ and $o - m = 30$, find $n + o$.
5. Suppose $j : k : l = 9 : 12 : 16$ with $l - j = 56$; compute $j + l$.

Tip: When solving problems with ratios, converting each term to a variable with a common multiple (e.g., $x = ak, y = bk, z = ck$) helps in setting up equations easily for further calculations.