SECTION 11.2 Systems of Linear Equations: Matrices 779 Financial Planning A newly married couple would like to visit Europe on their 20th anniversary. Since both work at home, they have decided to sell one of their cars and invest the money. To save enough money for the trip, they have determined they will need to earn interest of $250 annually. They have narrowed their investment choices down to Treasury notes that yield 3%,Treasury bonds that yield 5%, and corporate bonds that yield 6%. They have $6000 from the sale of their car to invest and want the amount invested in Treasury notes to equal the total amount invested in Treasury bonds and corporate bonds. How much should they place in each investment? Solution EXAMPLE 10 Let n b , , and c represent the amounts invested in Treasury notes,Treasury bonds, and corporate bonds, respectively.There is a total of $6000 to invest, which means that the sum of the amounts invested in Treasury notes, Treasury bonds, and corporate bonds should equal $6000. The first equation is + + = n b c 6000 (1) If $1000 is invested in Treasury notes, the annual income is ⋅ = 0.03 $1000 $30. In general, if n dollars are invested in Treasury notes, the annual income is n 0.03 . Since the total annual income is to be $250, the second equation is + + = n b c 0.03 0.05 0.06 250 (2) The amount invested in Treasury notes must equal the sum of the amounts invested in Treasury bonds and corporate bonds, so the third equation is = + − − = n b c n b c or 0 (3) We have the following system of equations: n b c n b c n b c 6000 0.03 0.05 0.06 250 0 + + = + + = − − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ (1) (2) (3) Begin with the augmented matrix, and use row operations to write the matrix in row echelon form. 1 1 1 0.03 0.05 0.06 1 1 1 6000 250 0 1 1 1 0 0.02 0.03 0 2 2 6000 70 6000 − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ → − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ↑ =− + R r r 0.03 2 1 2 =− + R r r 3 1 3 1 1 1 0 1 1.5 0 2 2 6000 3500 6000 1 1 1 0 1 1.5 0 0 1 6000 3500 1000 → − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ → ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ↑ ↑ = R r 1 0.02 2 2 = + R r r 2 3 2 3 The matrix is now in row echelon form. The final matrix represents the system n b c b c c 6000 1.5 3500 1000 + + = + = = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ (1) (2) (3) From equation (3), the couple should invest $1000 in corporate bonds. Back-substitute $1000 into equation (2) to find that = b 2000, so $2000 should be invested in Treasury bonds. Back-substitute these values into equation (1) and find that = n $3000, so $3000 should be invested in Treasury notes.
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