Chelsea took out three loans for a total of ?$78000 to start an organic orchard. Her?...

Chelsea took out three loans for a total of ?$78000 to start an organic orchard. Her? business-equipment loan was at an interest rate of? 11%, the? small-business loan was at an interest rate of 5?%, and her? home-equity loan was at an interest rate of 4.5?%. The total simple interest due on the loans in one year was ?$4605. The annual simple interest on the? home-equity loan was ?$105 more than the interest on the? business-equipment loan. How much did she borrow from each? source?

Expert Solution

Let the amounts borrowed by Chelsea as the business-equipment loan, the? small-business loan and the home-equity loan be $ x,$ y and $ z respectively. Since, the total amount of ? loans is $78000, hence x+y+z = 78000…(1). The amounts of interest, for 1 year, on the 3 loans are 0.11x, 0.05y and 0.045z respectively so that 0.11x+0.05y + 0.045z = 4605…(2). Also, the annual simple interest on the? home-equity loan was ?$105 more than the interest on the? business-equipment loan so that 0.045z = 0.11x + 105 or, -0.11x+0.045z = 105…(3).

The augmented matrix of this linear system is A =

1	1	1	78000
11/100	5/100	45/1000	4605
-11/100	0	45/1000	105

To solve the above linear system, we will reduce A to its RREF which is

1	0	0	15000
0	1	0	24000
0	0	1	39000

Hence x = 15000, y = 24000 and z = 39000. Thus, the amounts borrowed by Chelsea as the business-equipment loan, the? small-business loan and the home-equity loan were $ 15000,$ 24000 and $ 39000 respectively.

milcah answered 1 year ago

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