The given system of equations is:

$0 = 0, \quad 0 = 0, \quad 0 = 0$

Step-by-step analysis:

1. Each of these equations is always true for any values of the variables $x_1$, $x_2$, and $x_3$.
2. Since these equations impose no restrictions on $x_1$, $x_2$, and $x_3$, every possible combination of values for $x_1$, $x_2$, and $x_3$ will satisfy the system.

Conclusion:

The system is consistent and has infinitely many solutions.

Solution Set:

Since the system imposes no constraints, the solution set is the set of all points $(x_1, x_2, x_3) \in \mathbb{R}^3$. In set notation:

$\{ (x_1, x_2, x_3) \mid x_1, x_2, x_3 \in \mathbb{R} \}$

This represents all real numbers for the variables $x_1$, $x_2$, and $x_3$.

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Related Questions:

1. What does it mean for a system of equations to be inconsistent?
2. How can you tell if a system of linear equations has a unique solution?
3. What are the general methods to solve a system of linear equations?
4. How would this solution change if the system had specific equations, like $x_1 + x_2 = 0$?
5. What is the geometric interpretation of solutions in $\mathbb{R}^3$?

Tip:

A system with equations like $0 = 0$ is called indeterminate, meaning it holds for all values of the variables because these equations provide no information.