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To solve this problem, we need to evaluate the expression:

\[
x^2 - xz - xy + yz
\]

given that:

1. \( x - y = 5 \)
2. \( y - z = 3 \)

**Step 1: Express \( y \) and \( z \) in terms of \( x \)**

Since \( x - y = 5 \), we can rewrite \( y \) as:
\[
y = x - 5
\]

Similarly, since \( y - z = 3 \), we can express \( z \) in terms of \( y \) and thus \( x \):
\[
z = y - 3 = (x - 5) - 3 = x - 8
\]

So, we have:
\[
y = x - 5 \quad \text{and} \quad z = x - 8
\]

**Step 2: Substitute \( y \) and \( z \) into the expression**

Substitute \( y = x - 5 \) and \( z = x - 8 \) into the expression \( x^2 - xz - xy + yz \):

\[
x^2 - x(x - 8) - x(x - 5) + (x - 5)(x - 8)
\]

**Step 3: Expand each term**

1. \( x^2 \) remains as it is.
2. \( -x(x - 8) = -x^2 + 8x \)
3. \( -x(x - 5) = -x^2 + 5x \)
4. \( (x - 5)(x - 8) = x^2 - 8x - 5x + 40 = x^2 - 13x + 40 \)

**Step 4: Combine all terms**

Now, we combine these terms:
\[
x^2 - x^2 + 8x - x^2 + 5x + x^2 - 13x + 40
\]

Simplifying, we find:
\[
= 40
\]

Thus, the result is indeed:

\[
\boxed{40}
\]

Would you like additional steps explained or further assistance with similar problems?

Here are some related questions:
1. How would the solution change if \( x - y = 4 \) instead of 5?
2. What if \( y - z = 2 \)? How would that affect the result?
3. How could this problem be solved using matrix methods?
4. What if we had to solve for \( x \), \( y \), and \( z \) explicitly before substitution?
5. How can we use algebraic identities to simplify expressions like this in other cases?

**Tip:** When solving expressions with given conditions, express all variables in terms of one variable to simplify substitutions and calculations.

Given that x - y = 5 and y - z = 3, find the value of the expression x^2 - xz - xy + yz.

Learn how to evaluate the expression x^2 - xz - xy + yz given the conditions x - y = 5 and y - z = 3. This solution involves algebraic substitution and expression simplification, suitable for high school students in Grades 10-12.

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