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%.
x+y+z = 2000000
y = 4x
.07x + .085y + .095z = 169750
Wouldn't y=4z since "8.5% (y) is four times the amount borrowed at 9.5% (z)?
Let's set up the system of linear equations:
1. The total amount borrowed equation:
x + y + z = $2,000,000
2. The total interest earned equation:
0.07x + 0.085y + 0.095z = $169,750
3. The relationship between the amount borrowed at 8.5% and 9.5%:
y = 4z
So, the system of linear equations is:
x + y + z = $2,000,000
0.07x + 0.085y + 0.095z = $169,750
y = 4z
To set up a system of linear equations, we need to translate the given information into equations.
Let's start with the first equation. We know that the total annual interest is $169,750. We can express the interest earned from each borrowed amount and sum them up to get the total interest.
The interest earned from borrowing at 7% is x * 0.07.
The interest earned from borrowing at 8.5% is y * 0.085.
The interest earned from borrowing at 9.5% is z * 0.095.
So, the first equation is:
0.07x + 0.085y + 0.095z = 169,750.
Now, we are given that the amount borrowed at 8.5% (y) is four times the amount borrowed at 9.5% (z). Mathematically, this can be expressed as:
y = 4z.
Finally, we know that the total borrowed amount is $2,000,000. So, the second equation is:
x + y + z = 2,000,000.
To recap, we have two equations:
0.07x + 0.085y + 0.095z = 169,750,
x + y + z = 2,000,000.
These two equations form a system of linear equations that can be solved to find the values of x, y, and z.