Question

A museum borrows $2,000,000 at simple annual interest to purchase new exhibits. Let x represent the amount borrowed at 7%, y represent the amount borrowed at 8.5%, and z represent the amount borrowed at 9.5%. Set up a system of linear equations to determine how much is borrowed at each rate given that the total annual interest is $169,750 and the amount borrowed at 8.5% is four times the amount borrowed at 9.5%. You do not have to solve the system.

Answer 1

Explanation:

Total amount borrowed by the museum is $2000000

Let x represent the amount borrowed at 7%

Let y represent the amount borrowed at 8.5%

Let z represent the amount borrowed at 9.5%.

The formula for simple interest is

l = PRT/100

Where

I = interest

P = principal

R = rate

T = time

For a Pincipal of $x borrowed at a rate of 7% and the time is 1 year,

the simple interest would be expressed as

I = (x × 7 × 1)/100 = 0.07x

Fora Principal of $y borrowed at a rate of 8.5% and a duration of 1 year, the simple interest would be expressed as

I = (x × 8.5 × 1)/100 = 0.085y

For a Principal of $z borrowed at rate of 9.5% and a duration of 1 year,

the simple interest would be expressed as

I = (x × 9.5 × 1)/100 = 0.095z

The total annual interest is $169,750 and the amount borrowed at 8.5% is four times the amount borrowed at 9.5%. This means that

y = 4z

The total annual interest would be expressed as

169750 = 0.07x + 0.085×4z + 0.095z

169750 = 0.07x + 0.34z + 0.095z

169750 = 0.07x + 0.435z

The museum borrowed $2,000,000 at simple annual interest. It means that

x + y + z = 2000000

Since y = 4z , then

x + 4z + z = 2000000

x + 5z = 2000000

So the equations are

x + 5z = 2000000

0.07x + 0.435z = 169750