Let's solve this problem step by step, following the structure you provided.

Step 1: Obtain the system of linear equations

Let:

• $x$ be the amount invested at 5% interest.
• $y$ be the amount invested at 8% interest.
• $z$ be the amount invested at 9% interest.

From the problem, we are given three key pieces of information:

1. The total investment is RM10,000: $x + y + z = 10,000$
2. The total annual interest earned is RM770. This means: $0.05x + 0.08y + 0.09z = 770$
3. The amount invested at 9% is twice the amount invested at 5%: $z = 2x$

Thus, we have the following system of linear equations:

1) \quad & x + y + z = 10,000 \\ 2) \quad & 0.05x + 0.08y + 0.09z = 770 \\ 3) \quad & z = 2x \end{aligned}$$ ### Step 2: **Write down the matrix equation** We can represent the system of equations in matrix form as: $$\begin{pmatrix} 1 & 1 & 1 \\ 0.05 & 0.08 & 0.09 \\ -2 & 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 10,000 \\ 770 \\ 0 \end{pmatrix}$$ ### Step 3: **Use Gaussian elimination to solve** We now solve the system using Gaussian elimination. #### Step 1: Write the augmented matrix: $$\begin{pmatrix} 1 & 1 & 1 & | & 10,000 \\ 0.05 & 0.08 & 0.09 & | & 770 \\ -2 & 0 & 1 & | & 0 \end{pmatrix}$$ #### Step 2: Eliminate the first column for rows 2 and 3: - For row 2, subtract $$0.05 \times$$ row 1 from row 2: $$\text{Row 2} = \begin{pmatrix} 0 & 0.03 & 0.04 & | & 270 \end{pmatrix}$$ - For row 3, add $$2 \times$$ row 1 to row 3: $$\text{Row 3} = \begin{pmatrix} 0 & 2 & 3 & | & 20,000 \end{pmatrix}$$ The matrix is now: \[ \begin{pmatrix} 1 & 1 & 1 & | & 10,000 \\ 0 & 0.03 & 0.04 & | & 270 \\ 0 & 2 &