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.
Question content area bottom
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.
  
z equals x plus y
​(Type an equation. Do not include the​ $ symbol in your​ answer.)
Part 4
d. Solve the system of equations to find how much is invested in each property.
A.
There is one solution. The man invested ​$
  
enter your response here at 6.5​%, ​$
  
enter your response here at 9​%, and ​$
  
enter your response here at 8​%.
B.
There are infinitely many solutions. If the man invests​ $z at 8​%, then he invests ​$
  
enter your response here at 6.5​% and ​$
  
enter your response here at 9​%.
​(Type expressions using z as the​ variable.)
C.
There is no solution.

Answer 1

Let's break down the problem step by step using the information provided. We have three properties with investments denoted as:

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

From the information given, we've already established the following equations:

Part 1

a. The sum of the three investments: \[ x + y + z = 380000 \]

Part 2

b. The total returns from the investments equals $30,550: \[ 0.065x + 0.09y + 0.08z = 30550 \]

Part 3

c. The amount invested at 8% equals the sum of the investments at the other two rates: \[ z = x + y \]

Part 4

Next, we will solve this system of equations.

1. From Equation (c): Substitute \( z \) in terms of \( x \) and \( y \): \[ z = x + y \]

2. Substituting \( z \) in Equations (a) and (b):

   • Substitute \( z \) in Equation (a): \[ x + y + (x + y) = 380000 \] \[ 2x + 2y = 380000 \] \[ x + y = 190000 \quad \text{(Equation 4)} \]

   • Now substitute \( z = x + y \) into Equation (b): \[ 0.065x + 0.09y + 0.08(x + y) = 30550 \] \[ 0.065x + 0.09y + 0.08x + 0.08y = 30550 \] \[ (0.065 + 0.08)x + (0.09 + 0.08)y = 30550 \] \[ 0.145x + 0.17y = 30550 \quad \text{(Equation 5)} \]

3. Now, we can use Equation (4) to express \( y \) in terms of \( x \): \[ y = 190000 - x \]

4. Substituting \( y \) into Equation (5): \[ 0.145x + 0.17(190000 - x) = 30550 \] \[ 0.145x + 32300 - 0.17x = 30550 \] \[ -0.025x + 32300 = 30550 \] \[ -0.025x = 30550 - 32300 \] \[ -0.025x = -1750 \] \[ x = \frac{-1750}{-0.025} = 70000 \]

5. Using \( x \) to find \( y \): \[ y = 190000 - 70000 = 120000 \]

6. Now, using \( x \) and \( y \) to find \( z \): \[ z = x + y = 70000 + 120000 = 190000 \]

Final Results

The investments in each property are:

• Investment at 6.5%: \( x = 70000 \)
• Investment at 9%: \( y = 120000 \)
• Investment at 8%: \( z = 190000 \)

Conclusion

Thus, the answers are:

• A. There is one solution. The man invested $70,000 at 6.5%, $120,000 at 9%, and $190,000 at 8%.