Solve by Gauss Elimination with pivoting 𝑥+2𝑦−𝑧=9 5𝑥−𝑦+7𝑧=8 2𝑥−4𝑦−3𝑧=4

To solve the system of equations using **Gaussian Elimination with pivoting**, we can represent the given system of equations in matrix form as follows:

\[
\begin{aligned}
1x + 2y - 1z &= 9 \\
5x - 1y + 7z &= 8 \\
2x - 4y - 3z &= 4
\end{aligned}
\]

### Step 1: Write the system as an augmented matrix
\[
\begin{bmatrix}
1 & 2  & -1 & | & 9 \\
5 & -1 & 7  & | & 8 \\
2 & -4 & -3 & | & 4
\end{bmatrix}
\]

### Step 2: Apply Gaussian Elimination with partial pivoting
#### a) Partial Pivoting for Row 1 (Find the largest absolute value in the first column):
We compare \( |1|, |5|, |2| \), and see that \( 5 \) (row 2) is the largest. So, we swap row 1 and row 2:

\[
\begin{bmatrix}
5 & -1 & 7  & | & 8 \\
1 & 2  & -1 & | & 9 \\
2 & -4 & -3 & | & 4
\end{bmatrix}
\]

#### b) Eliminate below pivot (make the elements below the first pivot zero):
- Eliminate the first column entry in row 2: subtract \( \frac{1}{5} \) of row 1 from row 2
- Eliminate the first column entry in row 3: subtract \( \frac{2}{5} \) of row 1 from row 3

For row 2, we do:
\[
\text{Row 2} = \text{Row 2} - \frac{1}{5} \times \text{Row 1}
\]
For row 3, we do:
\[
\text{Row 3} = \text{Row 3} - \frac{2}{5} \times \text{Row 1}
\]

Let's calculate the resulting matrix:

I'll compute these eliminations and show the next steps.After performing the row operations, the augmented matrix is:

\[
\begin{bmatrix}
5  & -1  & 7   & | & 8 \\
0  & 2.2 & -2.4 & | & 7.4 \\
0  & -3.6 & -5.8 & | & 0.8
\end{bmatrix}
\]

### Step 3: Pivot the second column
Now, we apply **partial pivoting** to the second column. We compare the absolute values of the second column entries below the pivot (in row 2 and row 3), i.e., \( |2.2| \) and \( |3.6| \). The largest value is \( 3.6 \), so we swap rows 2 and 3:

\[
\begin{bmatrix}
5  & -1  & 7   & | & 8 \\
0  & -3.6 & -5.8 & | & 0.8 \\
0  & 2.2  & -2.4 & | & 7.4
\end{bmatrix}
\]

### Step 4: Eliminate below pivot
We now eliminate the entry below the pivot in the second column:
- For row 3, subtract \( \frac{2.2}{-3.6} \) times row 2 from row 3.

Let's compute the next elimination step.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can continue the Gaussian elimination process manually or wait until I can assist with calculations again.

Would you like me to walk you through the remaining steps verbally, or help with anything else?

Here are some questions you might consider next:

1. How is partial pivoting helpful in Gaussian elimination?
2. What happens if we don't use pivoting in Gaussian elimination?
3. How can we interpret the result of a system with infinite solutions using Gaussian elimination?
4. What are the conditions for a unique solution in a system of linear equations?
5. How does the back-substitution process work after Gaussian elimination?

**Tip**: Always check your pivot element before eliminating to ensure numerical stability in Gaussian elimination.

Solve by Gauss Elimination with pivoting: x + 2y − z = 9, 5x − y + 7z = 8, 2x − 4y − 3z = 4

Learn how to solve the system of equations x + 2y − z = 9, 5x − y + 7z = 8, 2x − 4y − 3z = 4 using Gaussian Elimination with partial pivoting. The detailed solution involves matrix representation and elimination steps, suitable for Grades 10-12 students studying linear algebra.

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