We are given the following system of equations:

1. $|x| + y = 10$
2. $x + y = 2$

We are tasked with finding the value of $x - y$.

Step-by-step solution:

1. Subtract equation 2 from equation 1:

   $(|x| + y) - (x + y) = 10 - 2$ Simplifying the left-hand side: $|x| + y - x - y = 8$ This simplifies to: $|x| - x = 8$

2. Consider two cases for $x$:

   • Case 1: $x \geq 0$
     If $x \geq 0$, then $|x| = x$. Substituting this into the equation $|x| - x = 8$ gives: $x - x = 8$ Which simplifies to: $0 = 8$ This is a contradiction, so $x$ cannot be non-negative.

   • Case 2: $x < 0$
     If $x < 0$, then $|x| = -x$. Substituting this into the equation $|x| - x = 8$ gives: $-x - x = 8$ Simplifying: $-2x = 8$ Solving for $x$: $x = -4$

3. Substitute $x = -4$ into equation 2 to find $y$:

   Using $x + y = 2$, substitute $x = -4$: $-4 + y = 2$ Solving for $y$: $y = 6$

4. Now, find $x - y$:

   We have $x = -4$ and $y = 6$, so: $x - y = -4 - 6 = -10$

Thus, the value of $x - y$ is $\boxed{-10}$.

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Would you like a more detailed explanation of any part of the process? Here are some related questions for further practice:

1. Solve for $x$ and $y$ in the system $|x| + y = 12$ and $x + y = 4$.
2. Given $|x| + y = 15$ and $x + y = 5$, find $x - y$.
3. How would the solution change if the equation $|x| + y = 10$ were replaced with $|x| - y = 10$?
4. Solve the system $|x| + y = 10$ and $x + y = -2$.
5. Solve for $x$ and $y$ in $|x| - y = 5$ and $x + y = 2$.

Tip: Always check whether your assumptions about the sign of $x$ (positive or negative) lead to a contradiction, as this can help eliminate impossible cases.