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The system of equations given is:

\[
\begin{aligned}
x_1 + x_3 &= 0 \\
x_2 - x_3 &= 0 \\
x_1 + x_2 &= 0
\end{aligned}
\]

We need to determine if the system has:
- A: One solution.
- B: More than one solution.
- C: No solution.

### Step 1: Express variables in terms of one another

From equation \(x_1 + x_3 = 0\), we have:
\[
x_1 = -x_3
\]

From equation \(x_2 - x_3 = 0\), we have:
\[
x_2 = x_3
\]

### Step 2: Substitute into the third equation

Substituting \(x_1 = -x_3\) and \(x_2 = x_3\) into the third equation \(x_1 + x_2 = 0\):
\[
-x_3 + x_3 = 0
\]
This holds true, meaning that all three equations are consistent.

### Step 3: Conclusion

Since there are three variables (\(x_1, x_2, x_3\)) and only two independent equations (the third equation doesn't add new information), the system is underdetermined and has **infinitely many solutions**. The general solution will depend on the free variable \(x_3\).

Thus, the correct answer is:
- **B: More than one solution.**

Would you like further details on this process or clarification on any step?

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Here are 5 related questions to explore further:
1. How do you solve systems of equations using matrix methods?
2. What does it mean for a system to be consistent or inconsistent?
3. How can you determine the number of solutions to a system of linear equations?
4. What role does the rank of a matrix play in solving linear systems?
5. How does Gaussian elimination help in solving systems of equations?

**Tip**: For underdetermined systems, always check for free variables by comparing the number of equations to the number of unknowns.

The following system of equations has: x1 + x3 = 0, x2 - x3 = 0, x1 + x2 = 0. Does the system have: A. One solution, B. More than one solution, C. No solution?

Discover how to solve a system of linear equations involving three variables: x1 + x3 = 0, x2 - x3 = 0, and x1 + x2 = 0. This detailed step-by-step solution explores underdetermined systems and why this system has infinitely many solutions. Suitable for advanced high school and undergraduate students.

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