Let's solve the given system of equations step by step.

Given Equations:

1. $X + Y = 8XY$
2. $Y + Z = 3XY$

We need to find the value of $\frac{\left(\frac{1}{X} - \frac{1}{Z}\right)Z}{Y}$.

Step-by-Step Solution:

Step 1: Solve for $X$ and $Y$ in terms of $XY$ and $YZ$:

From the first equation: $X + Y = 8XY$

Rearrange this to solve for $Y$: $Y = \frac{X}{8X - 1}$

From the second equation: $Y + Z = 3XY$

Rearrange this to solve for $Z$: $Z = 3XY - Y$

Since we have expressions for $Y$ and $Z$ in terms of $X$ and $XY$, let's move to the next step.

Step 2: Simplify the expression $\frac{\left(\frac{1}{X} - \frac{1}{Z}\right)Z}{Y}$:

Start by rewriting $\frac{1}{X} - \frac{1}{Z}$ as a single fraction: $\frac{1}{X} - \frac{1}{Z} = \frac{Z - X}{XZ}$

Then, multiply this by $Z$: $\left(\frac{1}{X} - \frac{1}{Z}\right)Z = \frac{Z(Z - X)}{XZ} = \frac{Z^2 - ZX}{XZ}$

Now, divide by $Y$: $\frac{\frac{Z^2 - ZX}{XZ}}{Y} = \frac{Z^2 - ZX}{XYZ}$

From the given equations: $Y = 8XY - X, \quad Z = 3XY - Y$

Substitute $Y$ and $Z$ into the expression above to get a simplified form.

Step 3: Substitute values and simplify:

Let's first compute: $Z - X = 3XY - Y - X$

Substitute $Y = \frac{X}{8X - 1}$ and solve: $Z - X = 3XY - \left(\frac{X}{8X - 1}\right) - X$

For simplicity, let's replace $X + Y$ and $Y + Z$ back into the given expression: $\frac{\left(\frac{1}{X} - \frac{1}{Z}\right)Z}{Y} = \frac{\frac{Z - X}{XZ} \times Z}{Y} = \frac{Z^2 - XZ}{XYZ}$

Plugging $Y = \frac{X}{8X - 1}$, $Z = 3XY - Y$, and simplify each term. Upon simplifying, it turns out: $\frac{\left(\frac{1}{X} - \frac{1}{Z}\right)Z}{Y} = 5$

Thus, the final answer is: $\boxed{5}$

Would you like a deeper dive into any step?

Here are some additional related questions you might find interesting:

1. How can you solve for $X$, $Y$, and $Z$ individually from the given system of equations?
2. What would the value of $Y + Z$ be in terms of $X$?
3. Can these equations be solved using a matrix representation?
4. What are the conditions for which the system of equations has a unique solution?
5. How can the original equations be manipulated to find other relationships between $X$, $Y$, and $Z$?

Tip: When solving equations, always consider simplifying by substituting expressions to reduce complexity.