Let's denote the amounts invested in the three investments as $x$, $y$, and $z$ respectively. The problem can be formulated using the following three equations:

1. Total Investment: $x + y + z = 65000$

2. Total Annual Income: $0.06x + 0.08y + 0.09z = 4800$

3. Income Relationship: $0.09z = 0.08y + 600$

Step 1: Set up the system of equations

From the third equation, we can rearrange it to express $z$ in terms of $y$: $0.09z - 0.08y = 600$

Thus, the system of equations is:

x + y + z &= 65000 \quad \text{(Equation 1)}\\ 0.06x + 0.08y + 0.09z &= 4800 \quad \text{(Equation 2)}\\ -0.08y + 0.09z &= 600 \quad \text{(Equation 3)} \end{aligned}$$ ### Step 2: Convert to matrix form The system can be written in matrix form as: $$\begin{pmatrix} 1 & 1 & 1 \\ 0.06 & 0.08 & 0.09 \\ 0 & -0.08 & 0.09 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 65000 \\ 4800 \\ 600 \end{pmatrix}$$ ### Step 3: Solve the system using matrix algebra Let $$A$$ be the coefficient matrix, $$\mathbf{x}$$ the vector of unknowns, and $$\mathbf{b}$$ the constant vector: $$A = \begin{pmatrix} 1 & 1 & 1 \\ 0.06 & 0.08 & 0.09 \\ 0 & -0.08 & 0.09 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 65000 \\ 4800 \\ 600 \end{pmatrix}$$ To solve for $$\mathbf{x}$$, we can compute $$\mathbf{x} = A^{-1} \mathbf{b}$$, where $$A^{-1}$$ is the inverse of matrix $$A$$. ### Step 4: Calculation I will now calculate the solution using matrix algebra.The amounts invested in each investment are: - **Investment 1** (at 6%): \$30,000 - **Investment 2** (at 8%): \$15,000 - **Investment 3** (at 9%): \$20,000 These amounts satisfy the conditions provided in the problem. Would you like further details or have any questions about the solution? Here are 5 related questions to explore: 1. How do you calculate the total income from all investments using the individual rates? 2. What is the impact of changing the interest rates on the distribution of investments? 3. How does the income relationship between the second and third investments affect the solution? 4. How can the solution be verified using substitution into the original equations? 5. What if the total investment was different—how would that change the matrix? **Tip:** When solving systems of equations, matrix algebra provides a powerful and systematic approach, especially for complex systems.