Consider the following systems of linear equations (all in three variables x1, x2, x3). Find if there is any solution (i.e. the system is consistent). If there are solutions, find them all and write the solution set. e) 0 = 0, 0 = 0, 0 = 0

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.

Explore a system of linear equations where 0 = 0 holds true for all variables. Understand why the system is consistent and discover how this leads to infinitely many solutions, with a detailed step-by-step analysis suitable for students in Grades 11-12 or early college.

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