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We are asked to determine if the system of linear equations is consistent and, if so, to find the general solution. The system of equations is:

\[
\begin{aligned}
x_1 - 2x_2 &= 3 \\
2x_1 - 7x_2 &= 0 \\
-5x_1 + 8x_2 &= 5
\end{aligned}
\]

### Step 1: Write the system in augmented matrix form
The system of equations can be expressed in augmented matrix form:

\[
\left[\begin{array}{cc|c}
1 & -2 & 3 \\
2 & -7 & 0 \\
-5 & 8 & 5
\end{array}\right]
\]

### Step 2: Perform row reduction to echelon form
We will now perform Gaussian elimination to determine if the system is consistent.

1. **Row 1** is fine as it is.
2. For **Row 2**, subtract 2 times Row 1 from Row 2 to eliminate the \( x_1 \) term in Row 2:
   \[
   R2 \rightarrow R2 - 2 \times R1 \quad \Rightarrow \quad (2 - 2(1), -7 - 2(-2), 0 - 2(3))
   \]
   The result:
   \[
   R2 = \left[ 0, -3, -6 \right]
   \]

3. For **Row 3**, add 5 times Row 1 to Row 3 to eliminate the \( x_1 \) term in Row 3:
   \[
   R3 \rightarrow R3 + 5 \times R1 \quad \Rightarrow \quad (-5 + 5(1), 8 + 5(-2), 5 + 5(3))
   \]
   The result:
   \[
   R3 = \left[ 0, -2, 20 \right]
   \]

After this step, the matrix is:
\[
\left[\begin{array}{cc|c}
1 & -2 & 3 \\
0 & -3 & -6 \\
0 & -2 & 20
\end{array}\right]
\]

### Step 3: Further row reduction
Next, let's focus on the second pivot. We will normalize Row 2 by dividing by \(-3\):
\[
R2 \rightarrow \frac{R2}{-3} \quad \Rightarrow \quad \left[ 0, 1, 2 \right]
\]

For **Row 3**, add 2 times Row 2 to Row 3 to eliminate the \( x_2 \) term:
\[
R3 \rightarrow R3 + 2 \times R2 \quad \Rightarrow \quad (0 + 2(0), -2 + 2(1), 20 + 2(2))
\]
The result:
\[
R3 = \left[ 0, 0, 24 \right]
\]

Now the matrix looks like this:
\[
\left[\begin{array}{cc|c}
1 & -2 & 3 \\
0 & 1 & 2 \\
0 & 0 & 24
\end{array}\right]
\]

### Step 4: Back-substitution
The last equation \( 0x_1 + 0x_2 = 24 \) leads to an inconsistency since \( 0 \neq 24 \). Therefore, the system of equations is **inconsistent**.

### Conclusion
The system is inconsistent, meaning there is **no solution**.

Would you like more details, or do you have any questions? Here are some related questions you might explore:

1. What does it mean for a system of linear equations to be consistent?
2. How does Gaussian elimination help in solving systems of equations?
3. Can a system of equations be inconsistent but still have a unique solution for some variables?
4. How can you use matrix methods like row reduction to solve systems of equations in different contexts?
5. What happens when a system of equations has more variables than equations?

**Tip:** Always check for consistency when solving systems of equations by looking at the augmented matrix after row reduction.

Determine if the following system of equations is consistent, and if so, find the general solution:

x1 - 2x2 = 3,
2x1 - 7x2 = 0,
-5x1 + 8x2 = 5.

Learn how to determine if a system of equations is consistent by using Gaussian elimination. This problem explores row reduction and consistency in systems of linear equations. Suitable for undergraduate students and advanced high school algebra.

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