Section 2.3 Systems of Linear Equations:

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Section 2.3 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 or equal to the number of variables, then the system has no solution or infinitely many solutions.

Ex. A system with no solution: Matrix Reduces to... Notice the false statement 0 = 1 The system is inconsistent and has NO solution.

Ex. A system with infinitely many solutions: Matrix Notice the row of zeros. Reduces to... So or If we let z = t then the solution is given by (2 – t, 1 – t, t)

Ex. A system with more equations than variables: Matrix Reduces to... Notice the false statement No Solution

Ex. A system with more variables than equations: Matrix Reduces to... So or Infinitely many solutions If we let z = s and w = t then the solution is given by (1 – 2s + t, –s + t, s, t)