Irfan invests a total of RM10,000 in three accounts, one paying 5% interest, another
paying 8% interest, and the third paying 9% interest. The annual interest earned on the  . Solve using matrix gaussian elimination method
three investments last year was RM770. The amount invested at 9% was twice the
amount invested at 5%.
1) Obtain a system of linear equations to represent the given informations.
2) Write down the matrix equation from the system of linear equations.
3) How much was invested at each rate?

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:
\[
\begin{aligned}
1) \quad & x + y + z = 10,000 \\
2) \quad & 0.05x + 0.08y + 0.09z = 770 \\
3) \quad & z = 2x
\end{aligned}
\]

### 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 &

Irfan invests a total of RM10,000 in three accounts, one paying 5% interest, another paying 8% interest, and the third paying 9% interest. The annual interest earned on the three investments last year was RM770. The amount invested at 9% was twice the amount invested at 5%. Solve using matrix Gaussian elimination method.

Learn how to solve a system of linear equations representing an investment problem, where RM10,000 is invested at different interest rates of 5%, 8%, and 9%. Use matrix Gaussian elimination to determine the amounts invested at each rate. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation