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Part 1
Use the​ Gauss-Jordan method to solve the following system of equations.
3x
plus
7y
minus
2z
equals
0
5x
minus
nothingy
plus
5z
equals
1
8x
plus
6y
plus
3z
equals
1

To solve the system of equations using the Gauss-Jordan method, we'll first represent the system as an augmented matrix and then perform row operations to bring it into reduced row echelon form (RREF). The system of equations is:

\[
\begin{aligned}
1. & \ 3x + 7y - 2z = 0 \\
2. & \ 5x + 0y + 5z = 1 \\
3. & \ 8x + 6y + 3z = 1
\end{aligned}
\]

### Step 1: Write the augmented matrix

\[
\begin{bmatrix}
3 & 7 & -2 & | & 0 \\
5 & 0 & 5 & | & 1 \\
8 & 6 & 3 & | & 1
\end{bmatrix}
\]

### Step 2: Perform row operations to get the matrix into RREF

I'll now perform the row operations necessary to solve this system.The system of equations has been transformed into the following reduced row echelon form (RREF):

\[
\begin{bmatrix}
1 & 0 & 0 & | & 0.08 \\
0 & 1 & 0 & | & 0 \\
0 & 0 & 1 & | & 0.12
\end{bmatrix}
\]

From this, we can conclude the solution to the system of equations:

\[
\begin{aligned}
x & = 0.08 \\
y & = 0 \\
z & = 0.12
\end{aligned}
\]

### Final Solution:
\[
x = 0.08, \quad y = 0, \quad z = 0.12
\]

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

Here are some related questions to deepen understanding:
1. How does the Gauss-Jordan method differ from Gaussian elimination?
2. What are the key steps in reducing a matrix to RREF?
3. How can row operations affect the solution of a system?
4. Can all systems of equations be solved using Gauss-Jordan elimination?
5. What are the advantages of using the Gauss-Jordan method?

**Tip:** Always check if the system is consistent or inconsistent before starting the Gauss-Jordan method.

Use the Gauss-Jordan method to solve the following system of equations: 3x + 7y - 2z = 0, 5x + 0y + 5z = 1, 8x + 6y + 3z = 1

Solve the system of equations 3x + 7y - 2z = 0, 5x + 0y + 5z = 1, 8x + 6y + 3z = 1 using the Gauss-Jordan elimination method. Follow step-by-step transformations of the augmented matrix into reduced row echelon form (RREF). Ideal for high school students in Grades 9-12.

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