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The problem presented in the image requires solving a system of linear equations using the given augmented matrix. Here is the step-by-step process to solve the system:

### Augmented Matrix
The augmented matrix given is:

\[
\left[\begin{array}{ccc|c}
2 & -7 & 0 & 5 \\
6 & 3 & 2 & 1 \\
\end{array}\right]
\]

This corresponds to the following system of linear equations:

\[
\begin{aligned}
2x_1 - 7x_2 + 0x_3 &= 5 \quad \text{(Equation 1)} \\
6x_1 + 3x_2 + 2x_3 &= 1 \quad \text{(Equation 2)}
\end{aligned}
\]

### Step 1: Simplify Equation 1
From Equation 1:
\[
2x_1 - 7x_2 = 5
\]
We can solve for \(x_1\):
\[
x_1 = \frac{5 + 7x_2}{2}
\]

### Step 2: Substitute \(x_1\) into Equation 2
Substitute \(x_1\) into the second equation:
\[
6\left(\frac{5 + 7x_2}{2}\right) + 3x_2 + 2x_3 = 1
\]

Simplify:
\[
3(5 + 7x_2) + 3x_2 + 2x_3 = 1
\]
\[
15 + 21x_2 + 3x_2 + 2x_3 = 1
\]
\[
24x_2 + 2x_3 = 1 - 15
\]
\[
24x_2 + 2x_3 = -14
\]

### Step 3: Further Simplify
Divide the entire equation by 2:
\[
12x_2 + x_3 = -7
\]

### Solution:
The system now looks like this:

\[
\begin{aligned}
x_1 &= \frac{5 + 7x_2}{2}, \\
12x_2 + x_3 &= -7.
\end{aligned}
\]

This system has \(x_2\) and \(x_3\) as free variables, meaning the system has infinitely many solutions.

### Parametric Form of the Solution:
Let \(x_2 = t\) (where \(t\) is a free parameter), then:
\[
x_1 = \frac{5 + 7t}{2},
\]
\[
x_3 = -7 - 12t.
\]

So the solution can be expressed as:
\[
(x_1, x_2, x_3) = \left(\frac{5 + 7t}{2}, t, -7 - 12t\right), \quad \text{where } t \in \mathbb{R}.
\]

### Final Answer:
The solution set is:
\[
\left(\frac{5 + 7t}{2}, t, -7 - 12t\right), \quad \text{where } t \in \mathbb{R}.
\]

---

Would you like any more details on this solution, or do you have any questions?

Here are five related questions for further exploration:
1. How can we determine if a system of linear equations has a unique solution, no solution, or infinitely many solutions?
2. What is the significance of the rank of a matrix in solving systems of equations?
3. How can the method of Gaussian elimination be applied to solve a system of linear equations?
4. What are the conditions under which a matrix is invertible, and how does that relate to solving linear systems?
5. How do we interpret the solution set geometrically when the system has free variables?

**Tip:** When dealing with linear systems, always consider simplifying the matrix using row operations before attempting to solve the equations directly. This can save time and reduce errors.

Solve the system of linear equations given by the augmented matrix [2 -7 | 5; 6 3 | 1].

Learn how to solve a system of linear equations represented by an augmented matrix. The problem involves finding the solution to a system with two equations and three variables, leading to infinitely many solutions. Suitable for undergraduate students studying linear algebra.

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