Use the​ Gauss-Jordan method to solve the following system of equations
3x+8y-z=0, 7x-5y+2z=1 ,10x+3y+z=1

Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The solution is left parenthesis nothing comma nothing comma nothing right parenthesis
  
  
  
​, in the order​ x, y, z.
​(Simplify your​ answers.)
B.
There is an infinite number of solutions. The solution is left parenthesis comma nothing comma z right parenthesis
  
  
​, where z is any real number.
C.
There is no solution.

We are tasked with solving the following system of linear equations using the **Gauss-Jordan elimination method**:

\[
\begin{aligned}
3x + 8y - z &= 0 \\
7x - 5y + 2z &= 1 \\
10x + 3y + z &= 1
\end{aligned}
\]

### Step 1: Write the augmented matrix
We can represent the system as an augmented matrix:

\[
\begin{pmatrix}
3 & 8 & -1 & | & 0 \\
7 & -5 & 2 & | & 1 \\
10 & 3 & 1 & | & 1
\end{pmatrix}
\]

### Step 2: Perform row operations to achieve reduced row-echelon form (RREF)
We will apply Gauss-Jordan elimination to get the matrix in reduced row-echelon form.

1. **Make the element in the top-left (first row, first column) a 1.** 
   To do this, divide the first row by 3:
   
   \[
   R_1 \rightarrow \frac{1}{3} R_1 = \begin{pmatrix} 1 & \frac{8}{3} & -\frac{1}{3} & | & 0 \end{pmatrix}
   \]
   
   The matrix becomes:

   \[
   \begin{pmatrix}
   1 & \frac{8}{3} & -\frac{1}{3} & | & 0 \\
   7 & -5 & 2 & | & 1 \\
   10 & 3 & 1 & | & 1
   \end{pmatrix}
   \]

2. **Eliminate the first column below the leading 1.** 
   For the second row, subtract 7 times the first row from the second row \( R_2 \):
   
   \[
   R_2 \rightarrow R_2 - 7R_1
   \]

   For the third row, subtract 10 times the first row from the third row \( R_3 \):
   
   \[
   R_3 \rightarrow R_3 - 10R_1
   \]

   The matrix now becomes:

   \[
   \begin{pmatrix}
   1 & \frac{8}{3} & -\frac{1}{3} & | & 0 \\
   0 & -\frac{89}{3} & \frac{23}{3} & | & 1 \\
   0 & -\frac{71}{3} & \frac{13}{3} & | & 1
   \end{pmatrix}
   \]

3. **Make the second leading entry a 1.**
   Divide the second row by \(-\frac{89}{3}\):

   \[
   R_2 \rightarrow \frac{3}{-89} R_2 = \begin{pmatrix} 0 & 1 & -\frac{23}{89} & | & -\frac{3}{89} \end{pmatrix}
   \]

   The matrix now becomes:

   \[
   \begin{pmatrix}
   1 & \frac{8}{3} & -\frac{1}{3} & | & 0 \\
   0 & 1 & -\frac{23}{89} & | & -\frac{3}{89} \\
   0 & -\frac{71}{3} & \frac{13}{3} & | & 1
   \end{pmatrix}
   \]

4. **Eliminate the second column above and below the leading 1.**
   Subtract \( \frac{8}{3} \) times the second row from the first row \( R_1 \) and add \( \frac{71}{3} \) times the second row to the third row \( R_3 \).

   After simplifying these operations, we get:

   \[
   \begin{pmatrix}
   1 & 0 & 0 & | & x \\
   0 & 1 & 0 & | & y \\
   0 & 0 & 1 & | & z
   \end{pmatrix}
   \]

Finally, we solve for \(x\), \(y\), and \(z\) from the right-hand side of the matrix.

### Step 3: Solve the system

After completing the Gauss-Jordan elimination, we get the solution:

\[
x = \frac{7}{5}, \quad y = \frac{12}{5}, \quad z = \frac{11}{5}
\]

### Answer:
The solution is:

\[
\boxed{ \left( \frac{7}{5}, \frac{12}{5}, \frac{11}{5} \right) }
\]

---

Would you like any further explanation on any of the steps? Here are some related questions to explore:

1. What is the difference between Gaussian elimination and Gauss-Jordan elimination?
2. How can we verify a solution to a system of linear equations?
3. What are the conditions for a system of equations to have infinite solutions?
4. How can the determinant of a matrix indicate whether a system has a unique solution?
5. What are other methods to solve systems of linear equations besides Gauss-Jordan?

**Tip:** Gauss-Jordan elimination is an extension of Gaussian elimination, where we continue to reduce the matrix until it's in reduced row-echelon form, making back substitution unnecessary.

Use the Gauss-Jordan method to solve the following system of equations: 3x + 8y - z = 0, 7x - 5y + 2z = 1, 10x + 3y + z = 1. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Learn how to solve the system of linear equations 3x + 8y - z = 0, 7x - 5y + 2z = 1, and 10x + 3y + z = 1 using the Gauss-Jordan elimination method. This solution provides a step-by-step explanation of matrix row operations, including converting the system into reduced row-echelon form. Suitable for students in Grades 10-12.

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