A man has ​$380 comma 000 invested in three rental properties. One property earns 6.5​% per year on the​ investment, the second earns 9​%, and the third earns 8​%. The total annual earnings from the three properties is $ 30 comma 550​, and the amount invested at 8​% equals the sum of the first two investments. Let x equal the investment at 6.5​%, y equal the investment at 9​%, and z represent the investment at 8​%.

Question

Complete parts​ (a) through​ (d) below.
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Part 1
a. Write an equation that represents the sum of the three investments.
  
x plus y plus z equals 380000
​(Type an equation. Do not include the​ $ symbol in your​ answer.)
Part 2
b. Write an equation that states that the sum of the returns from all three investments is $ 30 comma 550.
  
enter your response here
​(Type an equation. Do not include the​ $ symbol in your​ answer.)

Answer 1

Let's break down the equations one at a time.

Part 1: Equation for the sum of the three investments.
Given that the total amount invested is $380,000, we can write the equation as:
\[ x + y + z = 380000 \]

Part 2: Equation for the sum of the returns from all three investments.
To find the returns from each investment, we can calculate them based on their respective interest rates.

• The return from the investment at 6.5% is \(0.065x\).
• The return from the investment at 9% is \(0.09y\).
• The return from the investment at 8% is \(0.08z\).

According to the problem, the total annual earnings from these properties is $30,550. Therefore, we can write the equation for the total returns as:
\[ 0.065x + 0.09y + 0.08z = 30550 \]

Now, we have the two equations needed for parts (a) and (b):

1. \(x + y + z = 380000\)
2. \(0.065x + 0.09y + 0.08z = 30550\)