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The problem in the image asks you to find a complete solution to the system of linear equations, and then determine whether there is no solution, one solution, or infinitely many solutions. The system of equations is:

\[
\begin{align*}
x_1 + x_2 + x_3 & = 1 \\
x_1 + 2x_2 + 2x_3 & = 3 \\
x_1 + x_2 + x_3 & = 4 
\end{align*}
\]

### Step 1: Write the system in matrix form
The system can be represented as an augmented matrix:

\[
\begin{pmatrix}
1 & 1 & 1 & | & 1 \\
1 & 2 & 2 & | & 3 \\
1 & 1 & 1 & | & 4 
\end{pmatrix}
\]

### Step 2: Perform row operations to simplify the matrix
We aim to row-reduce the matrix to its simplest form (Row Echelon Form).

1. Subtract the first row from the second row:
   \[
   R2 = R2 - R1 \implies \begin{pmatrix} 1 & 1 & 1 & | & 1 \\ 0 & 1 & 1 & | & 2 \\ 1 & 1 & 1 & | & 4 \end{pmatrix}
   \]

2. Subtract the first row from the third row:
   \[
   R3 = R3 - R1 \implies \begin{pmatrix} 1 & 1 & 1 & | & 1 \\ 0 & 1 & 1 & | & 2 \\ 0 & 0 & 0 & | & 3 \end{pmatrix}
   \]

### Step 3: Analyze the row-reduced matrix
The row-reduced matrix is:

\[
\begin{pmatrix}
1 & 1 & 1 & | & 1 \\
0 & 1 & 1 & | & 2 \\
0 & 0 & 0 & | & 3 
\end{pmatrix}
\]

The third row translates to the equation \(0x_1 + 0x_2 + 0x_3 = 3\), which simplifies to \(0 = 3\). This is a contradiction, which indicates that there is **no solution** to this system of equations.

### Conclusion
Given that the system leads to a contradiction, the correct answer is:

**(C) There is no solution. The system is inconsistent.**

---

Would you like more details on any of the steps? 

Here are 5 additional questions you might explore:

1. What does it mean for a system of equations to be inconsistent?
2. How do you interpret an augmented matrix in the context of linear equations?
3. What are the steps to converting a system of linear equations to Row Echelon Form?
4. What would indicate a system has infinitely many solutions?
5. Can you provide an example of a system that has a unique solution?

**Tip:** When solving systems of equations, always check for consistency by fully reducing the matrix and interpreting the results correctly.

Solve the system of linear equations and determine if there is no solution, one solution, or infinitely many solutions:

x_1 + x_2 + x_3 = 1
x_1 + 2x_2 + 2x_3 = 3
x_1 + x_2 + x_3 = 4

Explore how to solve a system of linear equations and identify whether it has no solution, one solution, or infinitely many solutions. This example highlights inconsistent systems of equations leading to a contradiction. Suitable for Grades 10-12.

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