True or False? In Exercises 69 and 70, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. 69. (a) A system of one linear equation in two variables is always consistent. (b) A system of two linear equations in three variables is always consistent. (c) If a linear system is consistent, then it has infinitely many solutions.

True or False? In Exercises 69 and 70, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. 69. (a) A system of one linear equation in two variables is always consistent. (b) A system of two linear equations in three variables is always consistent. (c) If a linear system is consistent, then it has infinitely many solutions.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter1: Systems Of Linear Equations

Section1.1: Introduction To Systems Of Linear Equations

Problem 69E: True or False? In Exercises 69 and 70, determine whether each statement is true or false. If a...

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True or False? In Exercises 69 and 70, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a A system of one linear equation in two variables is always consistent. b A system of two linear equations in three variables is always consistent. c If a linear system is consistent, then it has infinitely many solutions.
True or False? In Exercises 69 and 70, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a A linear system can have exactly two solutions. b Two systems of linear equations are equivalent when they have the same solution set. c A system of three linear equations in two variables is always inconsistent.
Proof Prove each property of the system of linear equation in n variables Ax=b. a If rank(A)=rank([Ab])=n, then the system has a unique solution. b If rank(A)=rank([Ab])n, then the system has infinitely many solution. c If rank(A)rank([Ab]), then the system is inconsistent.
What is a system of linear equations in the variables x, y, and z?