SOLUTION: Mr Tui inherits a sum of $10,000 in Australia. He wants to invest this sum into three investments: Cooperate Bills, Cooperate Bonds and Treasury Bonds, with annual rates of 12%, 8%

Algebra ->  Matrices-and-determiminant -> SOLUTION: Mr Tui inherits a sum of $10,000 in Australia. He wants to invest this sum into three investments: Cooperate Bills, Cooperate Bonds and Treasury Bonds, with annual rates of 12%, 8%      Log On


   

Question 1098643: Mr Tui inherits a sum of $10,000 in Australia. He wants to invest this sum into three investments: Cooperate Bills, Cooperate Bonds and Treasury Bonds, with annual rates of 12%, 8% and 4% respectively. He wants his investment in Cooperate Bonds to be $500 more than the investment in Treasury Bonds. Use the Gauss Jordon Elimination method to find how much Mr Tui should allocate to each of his investments, in order to get an annual income of exactly $1,000.
Answer by greenestamps(13195) About Me  (Show Source):
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Let x be the amount in cooperate (corporate?) bills, y the amount in cooperate (?) bonds, and z the amount in treasury bonds.

(1) The sum of the three investments is $10,000: x%2By%2Bz+=+10000
(2) The income from the three is $1000: 0.12x%2B0.08y%2B0.04z+=+1000
(3) The amount in cooperate bonds is to be $500 more than the amount in treasury bonds: y-z+=+500

The beginning matrix corresponding to those three equations is
matrix%283%2C4%2C1%2C1%2C1%2C10000%2C.12%2C.08%2C.04%2C1000%2C0%2C1%2C-1%2C500%29

Since you are using Gauss Jordan elimination, and since the third row in this beginning matrix has 0 in the first column and 1 in the second (which is what we want on the second row), it is probably going to be easiest to switch rows 2 and 3. And to make things easier, let's multiply that row with the decimals by 25 to get rid of those decimals:
matrix%283%2C4%2C1%2C1%2C1%2C10000%2C0%2C1%2C-1%2C500%2C3%2C2%2C1%2C25000%29

The next thing we need is a 0 in row 3 column 1. To get that, replace row 3 with (3 times row 1 minus row 3):
matrix%283%2C4%2C1%2C1%2C1%2C10000%2C0%2C1%2C-1%2C500%2C0%2C1%2C2%2C5000%29

The first column is done. Now use the 1 in row 2 column 2 to get 0's in the other rows of column 2. Replace row 1 with (row 1 minus row 2); replace row 3 with (row 3 minus row 2):
matrix%283%2C4%2C1%2C0%2C2%2C9500%2C0%2C1%2C-1%2C500%2C0%2C0%2C3%2C4500%29

Next get the required 1 in row 3 column 3 by dividing row 3 by 3:
matrix%283%2C4%2C1%2C0%2C2%2C9500%2C0%2C1%2C-1%2C500%2C0%2C0%2C1%2C1500%29

Then use the 1 in column 3 of row 3 to get 0's in the other rows of column 3. Replace row 1 with (row 1 minus 2 times row 3); replace row 2 with (row 2 plus row 3):
matrix%283%2C4%2C1%2C0%2C0%2C6500%2C0%2C1%2C0%2C2000%2C0%2C0%2C1%2C1500%29


Mr. Tui needs to invest $6500 in cooperate bills, $2000 in cooperate bonds, and $1500 in treasury bonds.