Question

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​%.
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.
  
0.065 x plus 0.09 y plus 0.08 z equals 30550
​(Type an equation. Do not include the​ $ symbol in your​ answer.)
Part 3
c. Write an equation that states that the amount invested at 8​% equals the sum of the other two investments.
  
enter your response here
​(Type an equation. Do not include the​ $ symbol in your​ answer.)

Answer 1

Let's define the variables as you have mentioned:

- \( x \) = investment at 6.5%
- \( y \) = investment at 9%
- \( z \) = investment at 8%

Now, let's address parts (a), (b), and (c):

### Part 1
a. The equation that represents the sum of the three investments is:

\[
x + y + z = 380000
\]

### Part 2
b. The equation that states that the sum of the returns from all three investments is $30,550 is:

\[
0.065x + 0.09y + 0.08z = 30550
\]

### Part 3
c. The equation that states that the amount invested at 8% equals the sum of the first two investments is:

\[
z = x + y
\]

Now you have the three equations formatted appropriately based on the problem statement provided.