Linear Algebra: A Modern Introduction

Published by Cengage Learning
ISBN 10: 1285463242
ISBN 13: 978-1-28546-324-7

Chapter 1 - Vectors - 1.1 The Geometry and Algebra of Vectors - Exercises 1.1 - Page 17: 55

Answer

$X=1 \;\;\;and\;\;\; X=5$

Work Step by Step

The elements of $\;\mathbb{Z}_{8}\;$ are 0,1,2,3,4,5,6,7 So, if there is a solution for this equation, then x is one of these numbers. $If \;\;x=0\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;6.(0)+3=1 \; {\color{Red}\rightarrow } \;\; 0+3= 1\;\;{\color{Red} \Rightarrow }\;\;3\neq 1\\ so \;\;0 \;is \;not \;solution\\\\ If \;x=1\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;6.(1)+3=1 \; {\color{Red}\rightarrow } \;\; 6+3= 1\;\;{\color{Red} \Rightarrow }\;\;1= 1\\ so\; 1\; is\; SOLUTION.\\\\ If\;x=2\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;6.(2)+3=1 \; {\color{Red}\rightarrow } \;\; 4+3= 1\;\;{\color{Red} \Rightarrow }\;\;7\neq 1\\ so\; 2\; is\;not\; solution\\\\ If \;x=3\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;6.(3)+3=1 \; {\color{Red}\rightarrow } \;\; 2+3= 1\;\;{\color{Red} \Rightarrow }\;\;5\neq 1\\ so\; 3\; is\; not\; solution\\\\ If\; x=4\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;6.(4)+3=1 \; {\color{Red}\rightarrow } \;\; 0+3= 1\;\;{\color{Red} \Rightarrow }\;\;3\neq 1\\ so\; 4 \;is\; not\; solution\\\\ If\; x=5\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;6.(5)+3=1 \; {\color{Red}\rightarrow } \;\; 6+3= 1\;\;{\color{Red} \Rightarrow }\;\;1= 1\\ so\; 5 \;is\; SOLUTION\\\\ If\; x=6\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;6.(6)+3=1 \; {\color{Red}\rightarrow } \;\; 4+3= 1\;\;{\color{Red} \Rightarrow }\;\;7\neq 1\\ so\; 6 \;is\; not\; solution\\\\ If\; x=7\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;6.(7)+3=1 \; {\color{Red}\rightarrow } \;\; 2+3= 1\;\;{\color{Red} \Rightarrow }\;\;5\neq 1\\ so\; 7 \;is\; not\; solution\\\\ X=1 \;\;\;and\;\;\; X=5$
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