Mark O. answered 05/29/16
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Hi Bianca,
This problem is pretty involved. I will try to write matrix equations as best as I can in this editor.
First of all, the original $20,000 principal is divided into three different sub-principles, which we can call P1, P2 and P3. We eventually want to find P1, P2 and P3.
First off, we can write the equation
P1 + P2 + P3 = 20,000
We know that the simple interest I1 on P1 is 7%. In other words, I1 = 0.07P1. In the same manner, we can write I2 = 0.09P2 and I3 = 0.11P3. We also know that all of the interest values must sum to 1780. So, we can write
I1 + I2 + I3 = 1780
But, why introduce these new variables I1, I2 and I3. We know I1 = 0.07P1, I2 = 0.09P2 and I3 = 0.11P3. So, this last equation can be written as
0.07P1 + 0.09P2 + 0.11P3 = 1780
Finally, we are told that the second investment is $3000 less than the first, and the third investment is $1000 less than the first. The equations that describe this last statement are:
P2 = P1 - 3000
P3 = P1 - 1000
But, we can combine these last two equations to eliminate P1: P1 = P2 + 3000 = P3 + 1000. Or, in eliminating P1 from this altogether, we can write
P2 + 3000 = P3 + 1000
Or
P2 - P3 = -2000
So, we now have three equations and three unknowns and we can now write and solve a matrix equation:
P1 + P2 + P3 = 20,000
0.07P1 + 0.09P2 + 0.11P3 = 1780
P2 - P3 = -2000
As an augmented matrix, this is
1 1 1 | 20,000
0.07 0.09 0.11 | 1780
0 1 -1 | -2000
Multiply the second row by 100 for convenience, and we get
1 1 1 | 20,000
7 9 11 | 178,000
0 1 -1 | -2000
7 9 11 | 178,000
0 1 -1 | -2000
Multiply the top row by 7 and the middle row by -1.
7 7 7 | 140,000
-7 -9 -11 | -178,000
0 1 -1 | -2000
-7 -9 -11 | -178,000
0 1 -1 | -2000
Hold the first row constant and add it to the second row.
7 7 7 | 140,000
0 -2 -4 | -38,000
0 1 -1 | -2000
0 -2 -4 | -38,000
0 1 -1 | -2000
Divide the top row by 7 and the second row by 2.
1 1 1 | 20,000
0 -1 -2 | -19,000
0 1 -1 | -2000
0 -1 -2 | -19,000
0 1 -1 | -2000
Hold the middle row constant and add it to the top row and to the bottom row.
1 0 -1 | 1,000
0 -1 -2 | -19,000
0 0 -3 | -21,000
0 -1 -2 | -19,000
0 0 -3 | -21,000
Divide the bottom row by -3.
1 0 -1 | 1,000
0 -1 -2 | -19,000
0 0 1 | 7,000
0 -1 -2 | -19,000
0 0 1 | 7,000
Multiply the top row and the bottom row by 2.
2 0 -2 | 2,000
0 -1 -2 | -19,000
0 0 2 | 14,000
0 -1 -2 | -19,000
0 0 2 | 14,000
Hold the bottom row constant and add it to the top row and to the middle row.
2 0 0 | 16,000
0 -1 0 | -5,000
0 0 2 | 14,000
0 -1 0 | -5,000
0 0 2 | 14,000
Now, divide the top row by 2, the middle row by -1, and the bottom row by 2.
1 0 0 | 8,000
0 1 0 | 5,000
0 0 1 | 7,000
0 1 0 | 5,000
0 0 1 | 7,000
Now, we can just read off the answers.
P1 = $8000
P2 = $5000
P3 = $7000
This is our answer. We can check it. We see that the second investment is $3000 less than the first and that the third investment is $1000 less than the first. We can also calculate the principals:
0.07P1 + 0.09P2 + 0.11P3 = 560 + 450 + 770 = 1780