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The given system of equations is:

\[
\begin{cases}
3(x + 1) + 2(x + 2y) = -8 \\
4(x + 1) - (x + 2y) = -7
\end{cases}
\]

Let's solve this system step by step.

### Equation 1:
\[
3(x + 1) + 2(x + 2y) = -8
\]
Expanding:
\[
3x + 3 + 2x + 4y = -8
\]
Simplifying:
\[
5x + 4y = -11 \quad \text{(Equation 1)}
\]

### Equation 2:
\[
4(x + 1) - (x + 2y) = -7
\]
Expanding:
\[
4x + 4 - x - 2y = -7
\]
Simplifying:
\[
3x - 2y = -11 \quad \text{(Equation 2)}
\]

Now we have the system:
1. \( 5x + 4y = -11 \)
2. \( 3x - 2y = -11 \)

Let's solve this system.

### Solving by Substitution or Elimination:

Multiply Equation 2 by 2 to eliminate \(y\):
\[
2(3x - 2y) = 2(-11)
\]
\[
6x - 4y = -22 \quad \text{(Equation 3)}
\]

Now, add Equation 1 and Equation 3:
\[
(5x + 4y) + (6x - 4y) = -11 + (-22)
\]
\[
11x = -33
\]
\[
x = -3
\]

Substitute \(x = -3\) into Equation 2:
\[
3(-3) - 2y = -11
\]
\[
-9 - 2y = -11
\]
\[
-2y = -2
\]
\[
y = 1
\]

### The solution is \(x = -3\) and \(y = 1\).

Now, we are asked to calculate \(x^2 - xy + y^2\):
\[
x^2 - xy + y^2 = (-3)^2 - (-3)(1) + (1)^2
\]
\[
= 9 + 3 + 1 = 13
\]

So, the value is **13**.

---

Would you like any further details or clarification on this solution? Here are some related questions:

1. How do you solve a system of equations by substitution?
2. Can you solve this system using matrices or determinants?
3. What are other methods to solve systems of linear equations?
4. How does solving nonlinear equations differ from linear systems?
5. Can systems of equations have no solution or infinite solutions?

**Tip:** When solving systems of equations, try different methods (like substitution or elimination) to find the one that works best for you!

Given the system of equations: 3(x + 1) + 2(x + 2y) = -8 and 4(x + 1) - (x + 2y) = -7, find the value of x^2 - xy + y^2 where (x, y) are the solutions of the system.

Learn how to solve a system of linear equations using substitution and elimination methods to find values of x and y, and calculate x^2 - xy + y^2. This guide provides step-by-step instructions suitable for high school students.

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