An entrepreneur has $4,000,000 invested in three business ventures. One venture
ID: 3122617 • Letter: A
Question
An entrepreneur has $4,000,000 invested in three business ventures. One venture earns 7.5% per year on the investment, the second earns 8%, and the third earns 9%. The total annual earnings from the three ventures is $337,000, and the amount invested at 9% equals the sum invested in the other two ventures. Let x equal the investment at 7.5%, y equal the investment at and z represent the investment at 9%. a. Write a system of linear equations in the variables x, y and z. b. Rewrite the system of linear equations in the matrix form Ax = b, where A is 3 times 3 a coefficient matrix, x is a 3 times 1 column vector of unknowns and b a 3 times 1 column vectors of constant terms. c. Using the Inverse Method of Matrices, solve the system of three equations to determine the amount of money that should be invested in each venture.Explanation / Answer
Solution
Let x, y and z be respectively the amount invested at 7.5%, 8% and 9%.
Then, the given conditions translate into the following 3 equations:
x + y + z = 4000000 [Total investment]
0.075x + 0.08y + 0.09z = 337000 or
75x + 80y + 90z = 337000000 [Total annual earning]
z = x + y or - x - y + z = 0 [condition that investment at 9% must be equal to sum of other two investments].
To put the above in matrix form, let (3x3) matrix A = [ 1 1 1]
[75 80 90]
[-1 -1 1]
(3x1) column vector x be xT =(x y z) and
(3x1) column vector b be bT =(4000000 337000000 0).
Then, equation in matrix form is: Ax = b. ………… (1)
Back-up Theory
Pre-multiplying the above matrix equation (1) by A-1, we get, x = A-1b ………… (2)
A-1 = (adjA)/| A |…………………………………………………………………(3)
Now, to work out solution,
Using Excel Function, | A | = 10; adjA = [170 -2 10] and A-1 = [ 17 -0.2 1 ]
[-165 2 -15] [-16.5 0.2 -1.5]
[ 5 0 5] [ 0.5 0 0.5]
And the solution is: x = 6x105, y = = 14x105 and z = = 20x105 ANSWER