1. Systems of Linear Equations:
Underdetermined and Overdetermined systems
Theorem
If the number of equations is greater than or equal to the number of variables then the system has no solution, one solution, or infinitely many solutions.
If the number of equations is less than the number of variables, then the system has no solution or infinitely many solutions.
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2. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. A system with no solution:
Matrix
Notice the false statement 0 = 1
The system is inconsistent and has NO solution.
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3. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. A system with infinitely many solutions:
Matrix
Notice the row of zeros.
. . .
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4. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Example (Cont.) So or
If we let z = t then the solution is given by (2 – t, 1 – t, t)
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5. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. A system with more equations than variables:
Matrix
Notice the false statement 0 = 1.
No Solution.
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

6. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. A system with more variables than equations:
Matrix or So
Infinitely many solutions.
If we let z = s and w = t then the solution is given by
(1 + 2s + t, s + t, s, t)
. . .
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.