Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 1 - Systems of Linear Equations - 1.2 Gaussian Elimination and Gauss-Jordan Elimination - 1.2 Exercises - Page 23: 45

Answer

100 000 dollars at 9%, 250 000 dollars at 10%, and 150 000 dollars at 12%

Work Step by Step

Firstly, we are given that the total amount borrowed is 500 000 dollars, and a part of it was borrowed at 9%, another at 10%, and the rest at 12%. Let the 9%=x, 10%=y and 12%=z. The first equation is: $x+y+z=500 000$ Secondly, we have that the total annual interest was 52 000 dollars, and we know that it is made up of the interest gained from the 9%, 10%, and 12% interest loans. The second equation is therefore: $0.09x+0.1y+0.12z=52 000$ Thirdly, we are told that the amount borrowed at 10% is 2.5 times the amount borrowed at 9%, therefore the third equation can be written as: $2.5x-y=0$ The system of equations: $x +y +z =500 000$ $0.09x+0.1y+0.12z=52 000$ $2.5x -y =0$ Augmented matrix: (See attached image for the augmented matrix) Elementary row operations: (See the attached image for the row operations in the matrices) 1) $Row2×100$ $Row3×2$ 2) $Row2-(Row1×9)$ $Row3-(Row1×5)$ 4) $Row3+(Row2×7)$ 5) $Row3×\frac{1}{16}$ Rewrite the system of equations now in Row-Echlon form: $x+y+z=500 000$ $y+3z=700 000$ $z=150 000$ By solving for the variables using back substitution we get: $z=150 000$, $y=250 000$, and $x=100 000$ Solution: 100 000 dollars at 9%, 250 000 dollars at 10%, and 150 000 dollars at 12%
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