1. Copyright © Cengage Learning. All rights reserved. 2 SYSTEMS OF LINEAR EQUATIONS AND MATRICES Warm-Up: Read p. 70 Applied Example 1

2. Copyright © Cengage Learning. All rights reserved. 2.1 Systems of Linear Equations: An Introduction

3. 3 Solutions of Systems of Equations

4. 4 When dealing with systems of equations involving three variables, the system may have one and only one solution, infinitely many solutions, or no solutions. Figure 5 illustrates each of these possibilities.

5. 5 Solutions of Systems of Equations In Figure 5a, the three planes intersect at a point corresponding to the situation in which System (2) has a unique solution. A unique solution Figure 5(a)

6. 6 Solutions of Systems of Equations Figure 5b depicts a situation in which there are infinitely many solutions to the system. Here, the three planes intersect along a line, and the solutions are represented by the infinitely many points lying on this line. Infinitely many solutions Figure 5(b)

7. 7 Solutions of Systems of Equations In Figure 5c, the three planes are parallel and distinct, so there is no point common to all three planes; System (2) has no solution in this case. No solution Figure 5(c)

8. 8 Solutions of Systems of Equations We may interpret the solution(s) to a system comprising a finite number of such linear equations to be the point(s) of intersection of the hyperplanes defined by the equations that make up the system. As in the case of systems involving two or three variables, it can be shown that only three possibilities exist regarding the nature of the solution of such a system: (1) a unique solution, (2) infinitely many solutions, or (3) no solution.

9. 9 Classwork p. 74 #21, 25, 29