Explanation Given: Total amount of money that has to be invested by the manager is $1,000,000. Interest on each account is 6%, 8% and 10%. Total interest to be earned is $86,000. The amount invested at 10% is equal to the sum of the amount at 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 three accounts be x, y and z . As the total amount of money that has to be invested by the manager is $1,000,000, the first equation becomes x + y + z = 1 , 000 , 000 . The second equation is obtained as 0.06 x + 0.08 y + 0.1 z = 86 , 000 , as the interest on each account is 6%, 8% and 10% and total interest to be earned is $86,000. The third equation will be z = x + y that is − x − y + z = 0 , as the amount invested at 10% is equal to the sum of the amount invested at 6% and 8%. Hence, the system of equations becomes, { x + y + z = 1 , 000 , 000 0.06 x + 0.08 y + 0.1 z = 86 , 000 − 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 0.06 0.08 0.1 − 1 − 1 1 | = ( ( 1 ) [ ( 0.08 ) ⋅ 1 − ( 0.1 ) ⋅ ( − 1 ) ] − ( 1 ) [ ( 0

College Algebra in Context with Integrated Review and Worksheets Plus MyLab Math with Pearson eText-- Access Card Package

1st Edition

ISBN: 9780134380155

Author: Ronald J. Harshbarger, Lisa S. Yocco

Publisher: PEARSON

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Chapter 7.5, Problem 34E

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Chapter 7.5, Problem 33E

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College Algebra in Context with Integrated Review and Worksheets Plus MyLab Math with Pearson eText-- Access Card Package

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 account.

Solution Summary: The author calculates the amount of money that should be invested in each account, based on Cramer's rule.

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