College Algebra (6th Edition)

Published by Pearson
ISBN 10: 0-32178-228-3
ISBN 13: 978-0-32178-228-1

Chapter 6 - Matrices and Determinants - Concept and Vocabulary Check - Page 608: 3


infinitely many solutions

Work Step by Step

In a square system, if Gaussian elimination results in a matrix with a row with all 0s,the system has an infinite number of solutions (contains dependent equations). ------------- The last row represents the equation 0=0, which is true for any triplet (x,y.z). The other two equations will define the relationships between x, y, and z. There will be infinitely many solutions. This is how:. We take z to be any real number, and back-substitute into row 2, $z=t,\quad t\in \mathbb{R},$ $y=-1+10t$ Back substituting both into row 1, $x=2+(-1+10t)+2t=12t+1$ And the solution set is $\{(12t+1, 10t-1, t),\quad t\in \mathbb{R}\} $ (The system has as many solutions as $\mathbb{R}$ has numbers)
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