Use Gauss-Jordan row reduction to solve the given system of equations. (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer using the parameters x, y, z, and/or w.)
x	 − 	10y	 + 	z	 − 	6w	 = 	1
7x	 + 	5y	 + 	z	 + 	7w	 = 	2
8x	 − 	5y	 + 	2z	 + 	w	 = 	3
(x, y, z, w) = 

 

To solve the given system of equations using Gauss-Jordan elimination, we first need to represent the system as an augmented matrix and perform row reduction. The system of equations is:

\[
\begin{aligned}
x - 10y + z - 6w &= 1 \\
7x + 5y + z + 7w &= 2 \\
8x - 5y + 2z + w &= 3
\end{aligned}
\]

### Step 1: Write the augmented matrix
The augmented matrix representing this system is:

\[
\begin{bmatrix}
1 & -10 & 1  & -6 & 1 \\
7 & 5 & 1 & 7 & 2 \\
8 & -5 & 2 & 1 & 3
\end{bmatrix}
\]

### Step 2: Apply row operations to get the matrix into reduced row echelon form (RREF)
We'll use row operations to get leading 1's in each row and eliminate the other values in the columns of those leading 1's.

#### Row 1 operations
Row 1 is already set up nicely with a leading 1 in the first column, so we can use it to eliminate the values in the first column of rows 2 and 3. Perform the following row operations:

- \( R_2 \to R_2 - 7R_1 \)
- \( R_3 \to R_3 - 8R_1 \)

This gives us:

\[
\begin{bmatrix}
1 & -10 & 1  & -6 & 1 \\
0 & 75 & -6 & 49 & -5 \\
0 & 75 & -6 & 49 & -5
\end{bmatrix}
\]

#### Row 2 operations
Notice that rows 2 and 3 are the same, which indicates the system may be dependent. We can eliminate one of these rows (since they are identical).

We'll divide \( R_2 \) by 75 to make the leading coefficient 1:

\[
R_2 \to \frac{1}{75}R_2
\]

This gives:

\[
\begin{bmatrix}
1 & -10 & 1  & -6 & 1 \\
0 & 1 & -\frac{6}{75} & \frac{49}{75} & -\frac{5}{75} \\
0 & 75 & -6 & 49 & -5
\end{bmatrix}
\]

After canceling \(R_3\),

Use Gauss-Jordan row reduction to solve the given system of equations. (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer using the parameters x, y, z, and/or w.)
 x − 10y + z − 6w = 1 
 7x + 5y + z + 7w = 2 
 8x − 5y + 2z + w = 3
(x, y, z, w) = 

Learn how to solve the system of equations using Gauss-Jordan elimination. This step-by-step guide covers key linear algebra concepts such as matrix row operations, RREF, and dependency of systems. Suitable for students in Grades 11-12 or college-level linear algebra courses.

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