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 8%, 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 3x3 a coefficient matrix, x is a 3x1 column vector of unknowns and b a 3x1 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.