Pivot the system about the element in row 1, column 1. (Enter the augmented matrix with all cells in their original position.)

2	6	|4	
5	1	|2


To pivot the system about the element in row 1, column 1, we follow these steps:

1. Make the pivot element (in this case, 2 in row 1, column 1) equal to 1 by dividing the entire first row by 2.
2. Eliminate the other elements in the same column (in this case, eliminate 5 in row 2, column 1) by making the other elements in that column equal to 0.

### Step 1: Make the pivot element 1
We divide the first row by 2:

\[
\left[\begin{array}{cc|c}
2 & 6 & 4 \\
5 & 1 & 2 \\
\end{array}\right]
\]

Divide row 1 by 2:

\[
\left[\begin{array}{cc|c}
1 & 3 & 2 \\
5 & 1 & 2 \\
\end{array}\right]
\]

### Step 2: Eliminate the element in row 2, column 1
Now, we need to make the element in row 2, column 1 (which is 5) equal to 0 by subtracting 5 times row 1 from row 2.

\[
R_2 \rightarrow R_2 - 5R_1
\]

Perform the operation:

- Row 2: \( 5 - 5(1) = 0 \)
- Row 2, column 2: \( 1 - 5(3) = 1 - 15 = -14 \)
- Row 2, augmented part: \( 2 - 5(2) = 2 - 10 = -8 \)

The new augmented matrix is:

\[
\left[\begin{array}{cc|c}
1 & 3 & 2 \\
0 & -14 & -8 \\
\end{array}\right]
\]

This is the matrix after pivoting around the element in row 1, column 1.

---

Would you like further details or have any questions?

Here are some related questions for deeper understanding:

1. What is the purpose of pivoting in solving a system of linear equations?
2. How do you continue solving this system using Gaussian elimination after pivoting?
3. How does the elimination process work for larger systems of equations?
4. What would happen if the pivot element was zero? How would you proceed?
5. How can this process be applied to solve systems of equations with more than two rows?

**Tip:** Always verify your pivot element is non-zero before proceeding to avoid complications in the process!

Pivot the system about the element in row 1, column 1. (Enter the augmented matrix with all cells in their original position.)

2  6  | 4
5  1  | 2

This problem demonstrates how to pivot an augmented matrix around the element in row 1, column 1 using Gaussian elimination. The matrix is transformed step-by-step, starting with making the pivot element 1 and eliminating other elements in the same column. Ideal for students in Grades 9-12 studying linear algebra.

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