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This system of linear equations corresponds to the matrix equation:

\[
\begin{pmatrix}
1 & 0 & -3 & 0 \\
0 & 1 & -1 & 0 \\
1 & 4 & -4 & -1 \\
1 & 3 & 0 & -2
\end{pmatrix}
\begin{pmatrix}
a \\
b \\
c \\
d
\end{pmatrix}
=
\begin{pmatrix}
0 \\
0 \\
0 \\
0
\end{pmatrix}
\]

We can solve this system using Gaussian elimination (or row reduction). I'll walk through the elimination process:

### Step 1: Write the augmented matrix.
\[
\begin{pmatrix}
1 & 0 & -3 & 0 & | & 0 \\
0 & 1 & -1 & 0 & | & 0 \\
1 & 4 & -4 & -1 & | & 0 \\
1 & 3 & 0 & -2 & | & 0
\end{pmatrix}
\]

### Step 2: Perform row operations to reduce the matrix.

**Subtract row 1 from row 3 and row 4 to eliminate the first element in rows 3 and 4:**

- Row 3: \( R3 - R1 \)
\[
\begin{pmatrix}
1 & 4 & -4 & -1 & | & 0
\end{pmatrix} -
\begin{pmatrix}
1 & 0 & -3 & 0 & | & 0
\end{pmatrix}
= \begin{pmatrix}
0 & 4 & -1 & -1 & | & 0
\end{pmatrix}
\]

- Row 4: \( R4 - R1 \)
\[
\begin{pmatrix}
1 & 3 & 0 & -2 & | & 0
\end{pmatrix} -
\begin{pmatrix}
1 & 0 & -3 & 0 & | & 0
\end{pmatrix}
= \begin{pmatrix}
0 & 3 & 3 & -2 & | & 0
\end{pmatrix}
\]

The matrix now looks like this:
\[
\begin{pmatrix}
1 & 0 & -3 & 0 & | & 0 \\
0 & 1 & -1 & 0 & | & 0 \\
0 & 4 & -1 & -1 & | & 0 \\
0 & 3 & 3 & -2 & | & 0
\end{pmatrix}
\]

### Step 3: Eliminate the second column in row 4 by subtracting an appropriate multiple of row 2.

- Row 4: \( R4 - 3 \times R2 \)
\[
\begin{pmatrix}
0 & 3 & 3 & -2 & | & 0
\end{pmatrix} -
3 \times
\begin{pmatrix}
0 & 1 & -1 & 0 & | & 0
\end{pmatrix}
= \begin{pmatrix}
0 & 0 & 6 & -2 & | & 0
\end{pmatrix}
\]

The matrix is now:
\[
\begin{pmatrix}
1 & 0 & -3 & 0 & | & 0 \\
0 & 1 & -1 & 0 & | & 0 \\
0 & 4 & -1 & -1 & | & 0 \\
0 & 0 & 6 & -2 & | & 0
\end{pmatrix}
\]

### Step 4: Eliminate the third column in row 3 by subtracting \( \frac{2}{3} \times R4 \) from row 3 to make row 4 have a leading 1:

I will continue solving the matrix through

Solve the given system of linear equations using Gaussian elimination.

This problem involves solving a system of linear equations using Gaussian elimination. Learn how to apply row reduction techniques and write the system in augmented matrix form to simplify the process.

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