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Understanding Linear Systems - The Number of Variables and Equations

Learn about underdetermined and overdetermined systems of linear equations. Discover the Theorem: If equations ≥ variables, then system has no solution, one solution, or infinitely many solutions. If equations ≤ variables, system has no solution or infinitely many solutions. Examples provided for systems with no solution, infinitely many solutions, more equations, and more variables.

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Understanding Linear Systems - The Number of Variables and Equations

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  1. Section 2.3 Systems of Linear Equations: Underdetermined and Overdetermined Systems

  2. 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.

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

  4. 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)

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

  6. 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)

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