Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond for the given total investment and annual return.

Question

You invest in AAA-rated bonds, A-rated bonds, and B-rated bonds. The average yields are 4.5% on AAA bonds, 5% on A bonds, and 9% on B bonds. You invest twice as much in B bonds as in A bonds. Let x, y, and z represent the amounts invested in AAA, A, and B bonds, respectively.

x	+	y	+	z	=	(total investment)
0.045x	+	0.05y	+	0.09z	=	(annual return)

2y - z = 0

Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond for the given total investment and annual return. (Round your answers to the nearest cent.)

Total Investment- $12,000  Annual Return- $815

amount invested in AAA-bonds $____

amount invested in A-bonds $____

amount invested in B-bonds $____

Answer 1

.09x +.05y + .045z = 815

y - 2z = 0

x + y + z = 12000

.09 .05 .045 | 815

0 1 -2 | 0

1 1 1 | 12000

after row operations

1 0 0 | 5885

0 1 0 | 2038

0 0 1 | 4077

$5885 in AAA

$ 2038 in A

$ 4077 in B

rounded approximations, ignoring cents

or solve by substitution & elimination

.09x + .05y + .045z = 815

z =2y

.09x +.05y +.09y = 815

.09x +.14y = 815

x + y + z = 12000

x + y +2y = 12000

x +3y = 12000

.09x +.27y = .09(12000) = 1080

.27y -.14y = 1080-815

.13y = 265

y = 265/.13 =$2038.46 invested in A bond

z = 2y = $4076.92 invested in B bonds

x = 12000-y-z = $5884.62 in AAA bonds

check the answers:

.09(5884.62) + .05(2038.46) + .045(4076.92)

= 529.71 + 101.92 + 183.46

= 815.09