https://file.aiquickdraw.com/mathbot/file/172672158485847ugqr4x.jpg

The image shows an augmented matrix corresponding to a system of linear equations. Let's solve the system by row-reducing this matrix to find the general solution.

The augmented matrix is:

\[
\begin{bmatrix}
4 & -5 & 3 & 0 \\
12 & -15 & 9 & 0 \\
16 & -20 & 12 & 0
\end{bmatrix}
\]

I will perform Gaussian elimination to row-reduce this matrix and find the solution for the variables.It seems there was an issue during row reduction, indicating a degeneracy in the system. Upon closer inspection, the second row becomes a row of zeros, which suggests the system may have either infinitely many solutions or no solution. Let's re-examine the structure of the matrix to determine the solution more clearly.

The second row becoming zero indicates a dependent system, and we will need to continue manually to express the solution using free variables. Let's analyze:

1. Row 1: \( 4x_1 - 5x_2 + 3x_3 = 0 \)
2. Row 2: \( 0 = 0 \) (so no new information)
3. Row 3: Dependent on previous rows.

Thus, we can express \( x_1 \) in terms of free variables \( x_2 \) and \( x_3 \). I'll rewrite the solution accordingly.From the row-reduction analysis, we conclude the system is underdetermined, implying it has infinitely many solutions. The second row becomes zero, while the first row still contains useful information.

The equation from the first row is:

\[
4x_1 - 5x_2 + 3x_3 = 0
\]

Solving for \(x_1\):

\[
x_1 = \frac{5}{4}x_2 - \frac{3}{4}x_3
\]

Here, both \(x_2\) and \(x_3\) are free variables. Therefore, the general solution is:

\[
x_1 = \frac{5}{4}x_2 - \frac{3}{4}x_3, \quad x_2 = \text{free}, \quad x_3 = \text{free}
\]

### General solution:
\[
x_1 = \frac{5}{4}x_2 - \frac{3}{4}x_3, \quad x_2 = \text{free}, \quad x_3 = \text{free}
\]

Do you need more details, or have any questions?

Here are some related questions:
1. How do we determine the number of free variables in a system of equations?
2. What does it mean when a row in a matrix becomes all zeros?
3. How do we interpret the presence of free variables in terms of geometric solutions?
4. Can this system be represented as a parametric equation?
5. What would change if the augmented matrix had non-zero constants on the right side?

**Tip:** When performing Gaussian elimination, always simplify and eliminate one row at a time to avoid computational errors.

Find the general solution of the system whose augmented matrix is given below. Write 'free' for any free variable. If the system has no solution, type DNE for each variable.

\[\begin{bmatrix}
4 & -5 & 3 & 0 \\
12 & -15 & 9 & 0 \\
16 & -20 & 12 & 0
\end{bmatrix}\]

Explore the general solution for a system of linear equations using Gaussian elimination. Understand the implications of dependent systems and how to express variables in terms of free variables. Suitable for undergraduate-level students.

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