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Solve question one b using the gauss elimination method but not the  gauss Jordan elimination

To solve **Question 1(b)** using the **Gauss Elimination Method**, we are tasked with solving the following system of equations:

\[
\begin{aligned}
1. \quad & x_1 + 2x_2 + x_3 = 4 \\
2. \quad & 3x_1 - 4x_2 - 2x_3 = 2 \\
3. \quad & 5x_1 + 3x_2 + 5x_3 = -1
\end{aligned}
\]

### Step-by-Step Solution Using Gauss Elimination:

#### Step 1: Write the system as an augmented matrix.

\[
\begin{pmatrix}
1 & 2 & 1 & | & 4 \\
3 & -4 & -2 & | & 2 \\
5 & 3 & 5 & | & -1
\end{pmatrix}
\]

#### Step 2: Perform row operations to transform the matrix into upper triangular form.

1. **Eliminate** the elements below the pivot in the first column by making the entries below 1 (pivot in row 1, column 1) into zero:

   - \( R_2 = R_2 - 3R_1 \)
   - \( R_3 = R_3 - 5R_1 \)

   This gives the new matrix:

   \[
   \begin{pmatrix}
   1 & 2 & 1 & | & 4 \\
   0 & -10 & -5 & | & -10 \\
   0 & -7 & 0 & | & -21
   \end{pmatrix}
   \]

2. **Eliminate** the element below the pivot in the second column by making the entry in row 3, column 2 equal to zero:

   - \( R_3 = R_3 - \frac{7}{10} R_2 \)

   This gives the final upper triangular matrix:

   \[
   \begin{pmatrix}
   1 & 2 & 1 & | & 4 \\
   0 & -10 & -5 & | & -10 \\
   0 & 0 & 3.5 & | & -14
   \end{pmatrix}
   \]

#### Step 3: Perform back substitution to find the values of \( x_1, x_2, x_3 \).

1. From the third row: 
   \[
   3.5x_3 = -14 \quad \Rightarrow \quad x_3 = \frac{-14}{3.5} = -4
   \]

2. From the second row: 
   \[
   -10x_2 - 5(-4) = -10 \quad \Rightarrow \quad -10x_2 + 20 = -10 \quad \Rightarrow \quad -10x_2 = -30 \quad \Rightarrow \quad x_2 = 3
   \]

3. From the first row: 
   \[
   x_1 + 2(3) + (-4) = 4 \quad \Rightarrow \quad x_1 + 6 - 4 = 4 \quad \Rightarrow \quad x_1 + 2 = 4 \quad \Rightarrow \quad x_1 = 2
   \]

### Solution:
\[
x_1 = 2, \quad x_2 = 3, \quad x_3 = -4
\]

This is the solution using the **Gauss Elimination Method**. Do you have any questions or would you like further clarification on any of the steps?

---

### Next Steps:
Here are 5 questions that could help deepen your understanding of this method:

1. What is the difference between Gauss elimination and Gauss-Jordan elimination?
2. How does back substitution work in solving an upper triangular matrix?
3. How can pivoting improve the stability of the Gauss elimination process?
4. What happens if there is a zero in the pivot position during Gauss elimination?
5. How would you interpret the solution geometrically for this system of equations?

### Tip:
Always double-check your row operations and matrix entries during elimination to avoid mistakes that might affect the solution.

Solve question one b using the gauss elimination method but not the gauss Jordan elimination.

Learn how to solve the system of equations using the Gauss Elimination Method. We provide a detailed step-by-step process for solving the system x1 + 2x2 + x3 = 4, 3x1 − 4x2 − 2x3 = 2, 5x1 + 3x2 + 5x3 = −1. Understand core concepts like row operations, augmented matrix, and back substitution. Suitable for university-level students.

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