determine how much is borrowed at each rate given

Question

A natural history museum borrows $2,000,000 at simple annual interest to purchase new exhibits. Some of the money is borrowed at 7%, some at 8.5%, and some at 9.5%. Use a system of linear equations to determine how much is borrowed at each rate given that the total annual interest is $157,000 and the amount borrowed at 8.5% is four times the amount borrowed at 9.5%. Solve the system of linear equations using matrices.

Answer 1

X+Y+Z = 2000000

 

0.07X + 0.085Y + 0.095Z = 157000

 

Y = 4Z

 

Substituting Y=4Z into the first and second equations:

 

X + 5Z = 2000000

0.07X + 0.085(4Z) + 0.095Z = 157000

 

X = 2000000 - 5Z

0.07X + 0.435Z = 157000

 

 

0.07(2000000 - 5Z) + 0.435Z = 157000

 

140000 - 0.35Z + 0.435Z = 157000

 

0.085Z + 140000 = 157000

 

0.085Z = 17000

 

Z = 200000

 

Y = 800000

 

X = 1000000