Copyright © Cengage Learning. All rights reserved. 2 SYSTEMS OF LINEAR EQUATIONS AND MATRICES

Copyright © Cengage Learning. All rights reserved. 2.3 Systems of Linear Equations: Underdetermined and Overdetermined Systems

3 In this section, we look at systems that have infinitely many solutions and those that have no solution. We also study systems of linear equations in which the number of variables is not equal to the number of equations in the system.

4 Solution(s) of Linear Equations

5 The next example illustrates the situation in which a system of linear equations has infinitely many solutions.

6 Example 1 – A System of Equations with an Infinite Number of Solutions Solve the system of linear equations given by x + 2y = 4 3x + 6y = 12 Solution: Using the Gauss-Jordan elimination method, we obtain the following system of equivalent matrices: The last augmented matrix is in row-reduced form. (9)

7 Example 1 – Solution Interpreting it as a system of linear equations, we see that the given System (9) is equivalent to the single equation x + 2y = 4 or x = 4 – 2y If we assign a particular value to y—say, y = 0—we obtain x = 4, giving the solution (4, 0) to System (9). By setting y = 1, we obtain the solution (2, 1). In general, if we set y = t, where t represents some real number (called a parameter), we obtain the solution given by (4 – 2t, t). cont’d

8 Example 1 – Solution Since the parameter t may be any real number, we see that System (9) has infinitely many solutions. Geometrically, the solutions of System (9) lie on the line on the plane with equation x + 2y = 4. The two equations in the system have the same graph (straight line), which you can verify graphically. cont’d

9 Example 1 – A System of Equations with an Infinite Number of Solutions Solve the system of linear equations given by x + 2y – 3z = 4 3x - y - 2z = 12 2x + 3y – 5z = -3 Solution: Using the Gauss-Jordan elimination method, we obtain the following system of equivalent matrices: The last augmented matrix is in row-reduced form.

10 Example 1 – Solution Interpreting it as a system of linear equations, we see that the given System is equivalent to the single equation x – z = 0 or x = zand y – z = -1 or y = z - 1 If we set z = t, where t is a parameter, then the system has infinitely many solutions given by (t, t – 1, t). cont’d

11 Practice p. 98 Self-Check Exercises #1 & 2