91Ó°ÊÓ

When dealing with loans, breaking them down into individual variables makes the problem more manageable. In Chelsea's case:
- We use **\( x \)** to represent the amount borrowed for the business-equipment loan at **7% interest**.
- **\( y \)** is the amount for the small-business loan at **5% interest**.
- **\( z \)** stands for the home-equity loan at **3.2% interest**.
Naming these variables allows us to establish clear and concise equations for each aspect of the financial situation. This approach simplifies further calculations and helps us keep track of the amounts and their respective interest rates.

Calculating the interest on each loan is essential for solving the problem. In Chelsea's situation, the interest calculation for each loan involves multiplying the loan amount by the interest rate. We then combine these interest calculations to ensure the total matches the given total interest:
- For the business-equipment loan: **0.07\( x \)**.
- For the small-business loan: **0.05\( y \)**.
- For the home-equity loan: **0.032\( z \)**.
Adding these amounts gives us the total interest: **0.07\( x \) + 0.05\( y \) + 0.032\( z \) = 5040**.
These calculations let us see the relationship between the loan amounts and the total interest paid.

Equations like the ones above may initially look complex, but they can be simplified step by step. Let's take one of the established equations:
**\( z = \frac{0.07 x + 1190}{0.032} \)**.
Substituting this expression back into our other equations breaks the problem into smaller, more manageable parts and allows us to isolate variables:
- Substitute into the total loan equation: **\( x + y + \frac{0.07 x + 1190}{0.032} = 120,000 \)**.
Solving for **\( y \)** and then substituting back to find **\( x \)** and **\( z \)** becomes more straightforward. Simplifying equations by isolating variables can transform a seemingly daunting problem into a series of simpler steps, each leading us closer to the solution.