Set this up into a system of equations

Question

Can you please set this up into a system of equations that I can use to solve in Excel. 

 

"a couple who just had a baby want to start saving for their child's college education. They have $5000 to invest in three different plans. A traditional savings account pays 3% interest, a certificate of deposit pays 6% annual interest, a a prepaid college plan pays 7.5% annual interest. If they want to invest the same amount in the prepaid college fund as in the other two plans together, how much should they invest in each plan to realize an interest income of exactly $300 for the first year?

Answer 1

First, Let x=$ invested at 3%

              y=$ invested at 6%

              z=$ invested at 7.5%

 

Next, set up a matrix with the following information:

 

1      x          +              1        y          +            1         z          =           5000

0.03    x           +          0.06    y          +           0.075   z          =             300

1        x            +         1         y          -            1          z          =                0

 

Which gives us the Iteration 0 of:

 

1          1          1          =          5000

0.03     0.06      0.075   =            300

1          1         -1          =               0

 

Next, perform the following row operations, making sure to update the matrix at each iteration:

 

[1]  -  R1 +      R3 = new R3

[2]  -3R1 + 100R2 = new R2

[3]      R3 +    2R1 = new R1

[4]   4.5R3 +  2R2 = new R2

[5]     -R2 +   3R1 = new R1

 

This gives the following Iteration 5 matrix:

 

6          0          0          =          7500

0          6          0          =          7500

0          0         -2          =         -5000

 

To which I perform the following row reductions:

 

 (1/6)R1 = new R1

 (1/6)R2 = new R2

(-1/2)R3 = new R3

 

Which gives us the following matrix:

 

1          0          0          =          1250

0          1          0          =          1250

0          0          1          =          2500

 

So, (x,y,z) is:  (1250,1250,2500)

 

Which means:  $1,250 invested in the savings account

                       $1,250 invested in the CD

                       $2,500 invested in the college savings plan

 

This yields the following checks:

 

[1]  1,250 + 1,250 + 2,500 = $5,000     [Check]

[2]  0.03 * 1250 + 0.06 * 1250 + 0.075 * 2500 = $300     [Check]

[3]  $1,250 + $1,250 - $2,500 = 0          [Check]

 

 

Answer 2

Hi Andrea:

 

First off, we can use I = prt since we are looking at a single year.

If we let x = amount invested 3% ... interest .03x

Let y = amount invested 6%... interest .06x

Let z = amount invested 7.5% ... (x + y) ... interest .075(x+y)

 

300 = 0.03x + 0.06y + 0.075(x+y)

300 = 0.03x + 0.06y + 0.075x + 0.075y

300 = 0.105x + 0.135y   {equation 1}

 

x + y + z = 5000

x + y + x + y = 5000

2x + 2y = 5000       {equation 2}

 

Questions?