Question

Set up the system of equations and then solve it by using an inverse matrix.

A trust account manager has $2,000,000 to be invested in three different accounts. The accounts pay 6%, 8%, and 10%, and the goal is to earn $172,000 with the amount invested at 10% equal to the sum of the other two investments. To accomplish this, assume that x dollars are invested at 8%, y dollars at 10%, and z dollars at 6%.
Find how much should be invested in each account to satisfy the conditions.
$ 6% rate
$ 8% rate
$ 10% rate

Answer 1

x=400000\\y=600000\\z=1000000

Explanation:

Given that a trust account manager has $2,000,000 to be invested in three different accounts. The accounts pay 6%, 8%, and 10%, and the goal is to earn $172,000 with the amount invested at 10% equal to the sum of the other two investments.

To accomplish this, assume that x dollars are invested at 8%, y dollars at 10%, and z dollars at 6%.

The equations formed would be

`x+y+z= 2000000`

Interest amount =
`6x+8y+10z =172000(100)`

`z=x+y\\x+y-z=0`

these three can be written in matrix form as

`\left[\begin{array}{ccc}1&1&1\\6&8&10\\1&1&-1\end{array}\right] =\left[\begin{array}{ccc}200000\\17200000\\0\end{array}\right]`

The inverse of the matrix is

`\left[\begin{array}{ccc}9/2&-1/2&-1/2\\-4&1/2&1\\1/2&0&-1/2\end{array}\right]`

X = A inverse *B

Using this we get

`x=400000\\y=600000\\z=1000000`