Explanation Given: A woman has $750,000 invested in three rental properties. One yields an annual return of 12% on her investment, and second returns 15% per year on her investment and the third investment returns 10%. Definition used: 1. Any augmented matrix that has 1`s or 0`s on the diagonal of its coefficient part and 0`s below the diagonal is said to be in row-echelon form. 2. The augmented matrix representing n equations in n variables is said to be in reduced row-echelon form if it has 1`s or 0`s on the diagonal of its coefficient part and 0`s everywhere else. Calculation: Let x represent the amount invested in rental property that earns 12%, y represent the amount invested in rental property that earns 15% and z represent the amount invested in rental property that earns 10%. The total money invested is $750,000. Thus, x + y + z = 750 , 000 . Given that, the total annual return from the three investments is $89,500. Thus, the equation that represents the total earning by the three investments is, 0.12 x + 0.15 y + 0.10 z = 89 , 500 Given that, the amount invested in 10% investment is limited to 1 2 the sum of the other two investments. That is, z = 1 2 ( x + y ) . Combine the above three equations and form the system of three equations in three variables x , y and z as follows. { x + y + z = 750 , 000 0.12 x + 0.15 y + 0.10 z = 89 , 500 z = 1 2 ( x + y ) Simplify the above system and rewrite as follows. x + y + z = 750 , 000 (1) 0.12 x + 0.15 y + 0.10 z = 89 , 500 (2) x + y − 2 z = 0 (3) The augmented matrix associated with the system of equations in x , y and z is, [ 1 1 1 750 , 000 0.12 0.15 0.10 89 , 500 1 1 − 2 0 ] Observe that the augmented matrix is not in row-echelon form. Hence apply certain row operations and reduce into row-echelon form as follows. Perform the row operations R 2 − 0.12 R 1 → R 2 , and R 3 − R 1 → R 3 and obtain the new row reduced matrix as, [ 1 1 1 750 , 000 0 0

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Chapter 7, Problem 37RE

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Chapter 7 Solutions

EBK COLLEGE ALGEBRA IN CONTEXT

Ch. 7.1 - In Exercises 14, solve the systems of equations....Ch. 7.1 - Prob. 2E Ch. 7.1 - Prob. 3E Ch. 7.1 - Prob. 4E Ch. 7.1 - Prob. 5E Ch. 7.1 - Prob. 6E Ch. 7.1 - Prob. 7E Ch. 7.1 - Prob. 8E Ch. 7.1 - Prob. 9E Ch. 7.1 - Prob. 10E

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Solution Summary: The author explains how a woman invests 750,000 in three rental properties. One yields an annual return of 12%, the second returns 15%, and the third returns 10%.

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