Explanation Given: Total amount of money that has to be invested by a woman is $800,000. Profit earned on each property’s share is 9%, 8% and 7.5%. Total yearly earnings is $67,400. The amount invested on property which earns 9% is equal to the sum of the amount invested on properties which earns 6% and 8%. Definition used: Cramer’s rule: The solution to { a 1 x + b 1 y + c 1 z = d 1 a 2 x + b 2 y + c 2 z = d 2 a 3 x + b 3 y + c 3 z = d 3 is x = D x D , y = D y D and z = D z D , where D = | a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 | , D = | a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 | , D = | a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 | , D = | a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 | D ≠ 0 . If D = 0 and at least one of the determinants in the numerators, D x , D y and D z is not zero, then the system is inconsistent and the system has no solution. If D = 0 and all the determinants in the numerators, D x , D y and D z are zero, then the system of equations is dependent and the system has infinitely many solutions. Formula used: The determinant of the matrix [ a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ] is det [ a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ] = | a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 | = a 1 b 2 c 3 + b 1 c 2 a 3 + c 1 a 2 b 3 − a 3 b 2 c 1 − b 3 c 2 a 1 − c 3 a 2 b 1 . Calculation: Let the amount invested in property which earns profit at 9%, 8% and 7.5% be x, y and z , respectively. As the total amount of money that has to be invested by the woman is $800,000, the first equation becomes x + y + z = 800 , 000 . The second equation is obtained as 0.09 x + 0.08 y + 0.075 z = 67 , 400 , as the profit earned on each property’s share is 9%, 8% and 7.5% and total yearly earnings is $67,400. The third equation will be x = y + z that is x − y − z = 0 as the amount invested on property which earns 9% is equal to the sum of the amount invested on properties which earns 6% and 8%. Hence, the system of equations becomes, { x + y + z = 800 , 000 0.09 x + 0.08 y + 0.075 z = 67 , 400 x − y − z = 0 Solve the system of equations as follows. By above formula, compute the determinant of the 3 × 3 matrix. D = | 1 1 1 0.09 0.08 0.075 1 − 1 − 1 | = ( ( 1 ) [ ( 0.08 ) ⋅ ( − 1 ) − ( 0.075 ) ⋅ ( − 1 ) ] − ( 1 ) [ ( 0.09 ) ⋅ ( − 1 ) − ( 0.075 ) 1 ] + ( 1 ) [ ( 0.09 ) ( − 1 ) − ( 0

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Chapter 7.5, Problem 37E

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Chapter 7.5, Problem 36E

Chapter 7.5, Problem 38E

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EBK COLLEGE ALG.IN CONTEXT W/APPL.F/MAN

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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The amount of money that should be invested in each property.

Solution Summary: The author explains Cramer's rule: the amount of money that should be invested in each property is 400,000, 280,000 and 120,000 respectively.

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