Question

I need help to understand how to formulate this as equations and find 2 solutions:

Mr. and Mrs. Garcia have a total of $100,000 to be invested in stocks, bonds, and a money market account. The stocks have a rate of return of 18%/year, while the bonds and the money market account pay 12%/year and 6%/year, respectively. The Garcias have stipulated that the amount invested in stocks should be equal to the sum of the amount invested in bonds and 3 times the amount invested in the money market account.
 
How should the Garcias allocate their resources if they require an annual income of $15,000 from their investments? Give two specific options. (Let x1, y1, and z1 refer to one option for investing money in stocks, bonds, and the money market account respectively. Let x2, y2, and z2 refer to a second option for investing money in stocks, bonds, and the money market account respectively.)

Accepted Answer

Use the information given to set up equations as follows:
Assign variables: Let x be the amount invested in stocks, y the amount invested in bonds, and z the amount invested in money markets.
(1) The total amount invested is 100,000 which is the sum of the amounts invested:

x+y+z=100,000

(2) The total amount of income from the investments is 15,000. This is the sum of the incomes from the three investments. Since stocks earn 18% the income from stocks is .18x; similarly the income from bonds is .12y and from money markets is .06z.
So .18x+.12y+.06z=15,000 Multiplying through by 100 (to ease computations) we get:

18x+12y+6z=1,500,000

Now divide by 6:

3x+2y+z=250,000

(3) The amount invested in stocks is to be the sum of the amount invested in bonds and three times the amount invested in money markets. So x=y+3z. Rewriting in standard form we get:

x-y-3z=0

The system of equations:

x+y+z=100,0003x+2y+z=250,000x-y-3z=0

There are a number of methods to solve this system. Using Gaussian elimination we get:

([1,1,1,|,100000],[3,2,1,|,250000],[1,-1,-3,|,0])==>([1,0,-1,|,50000],[0,1,2,|,50000],[0,0,0,|,0])

The last row of zeros indicates that the system is dependent (there are many solutions.)
We see that x-z=50000 and y+2z=50000. Letting z be the independent variable the solutions can be described as:

(50000+z, 50000-2z, z)

For example we can let z=10,000; then x=60,000 and y=30,000. Note that the sum is 100,000 and .18x+.12y+06z=15,000.
Another choice might be z=20,000; then x=70,000 and y=10,000.

Math

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