https://file.aiquickdraw.com/mathbot/file/1726500041695c89xjjcy.jpg

To determine how many solutions this system of equations has, let's analyze the system step by step.

The system is:

\[
\begin{aligned}
x_1 + x_3 &= 1 \\
x_2 + x_3 &= 1 \\
x_1 + x_2 &= 0
\end{aligned}
\]

### Step 1: Express \(x_1\) and \(x_2\) in terms of \(x_3\).

- From the first equation: 
  \[
  x_1 = 1 - x_3
  \]

- From the second equation: 
  \[
  x_2 = 1 - x_3
  \]

### Step 2: Substitute \(x_1 = 1 - x_3\) and \(x_2 = 1 - x_3\) into the third equation.

The third equation is:
\[
x_1 + x_2 = 0
\]

Substitute for \(x_1\) and \(x_2\):

\[
(1 - x_3) + (1 - x_3) = 0
\]

Simplifying:
\[
2 - 2x_3 = 0
\]

Solve for \(x_3\):
\[
2x_3 = 2 \quad \Rightarrow \quad x_3 = 1
\]

### Step 3: Substitute \(x_3 = 1\) into the expressions for \(x_1\) and \(x_2\).

- \(x_1 = 1 - 1 = 0\)
- \(x_2 = 1 - 1 = 0\)

### Conclusion:

The solution to the system is:
\[
x_1 = 0, \quad x_2 = 0, \quad x_3 = 1
\]

This is a unique solution, meaning **the system has one solution**.

Thus, the correct answer is **A: One solution**.

---

Would you like more details or have any further questions?

Here are some related questions:
1. How do you check if a system of linear equations is consistent?
2. What methods can be used to solve a system of linear equations?
3. How do you interpret the rank of a matrix in terms of solutions of a system?
4. What are the conditions for a system of equations to have no solution?
5. How can we identify if a system of equations has infinitely many solutions?

**Tip:** For larger systems, using methods like Gaussian elimination or matrix representations can make solving easier.

The following system of equations has:

x_1 + x_3 = 1
x_2 + x_3 = 1
x_1 + x_2 = 0

A: One solution. B: More than one solution. C: No solution.

This system of linear equations: x_1 + x_3 = 1, x_2 + x_3 = 1, and x_1 + x_2 = 0 is solved using the substitution method. Learn step-by-step how to find the unique solution, where x_1 = 0, x_2 = 0, and x_3 = 1. Suitable for high school students.

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