College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 6, Matrices and Determinants - Section 6.1 - Matrices and Systems of Linear Equations - 6.1 Exercises - Page 500: 3


a.$\quad$ x and y b.$\quad$ dependent c.$\quad$ $x=3+z,\ y=5-2z,\ z=$ any real number.

Work Step by Step

a. Leading variables correspond to leading entries in the reduced row-echelon form. x and y have a corresponding leading 1 in their correspoding columns, z does not. Answer: x and y. b. Since the last row represents an equation 0=0, which is always true, the system will have solutions. Since not all variables are leading, it will be dependent. Answer: dependent. c. Taking z to be any real number (since it has no corresponding leading entry), we back-substitute to find x and y. $x-z=3\quad\Rightarrow\quad x=3+z$ $y+2z=5\quad\Rightarrow\quad y=5-2z$ Answer: $x=3+z,\ y=5-2z,\ z=$ any real number.
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