Solving Linear Equations in Systems
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Questions and Answers

What is the correct value of $z$ from the given system of equations?

  • 4
  • 2 (correct)
  • 1
  • 3

    • Which equation corresponds to the second row after performing row operations?

  • $y + 3z = 5$ (correct)
  • $-3y + z = -5$
  • $-y + 3z = 5$
  • $0y + 2z = 6$

    • What equation must be solved to find the value of $y$?

  • $z - 2y = 1$
  • $2z = 4$
  • $x - 2y + 9 = 0$
  • $y + 6 = 5$ (correct)

    • What transformation was performed on the first row during Gaussian elimination?

    <p>Addition of $R_1$ to $R_2$</p> Signup and view all the answers

    What is the final value of $x$ after solving the equations?

    <p>1</p> Signup and view all the answers

    Study Notes

    Linear Equations Systems

    • Systems of linear equations can have one solution, no solution, or infinitely many solutions.
    • Methods to solve systems of linear equations include Gaussian elimination and substitution.

    Solving Systems of Equations (Examples)

    • Example 1:

      • x + y = 5
      • y = 5 – x
      • This system has infinitely many solutions, expressed parametrically as p = t; y = 5 – t.

    • Example 2:

      • x + y = 2
      • -x – y = 3
      • This system has no solution.

    • Example 3:

      • x + y = 2
      • x – y = 4
      • Solution: x = 3; y = -1

    • Example 4:

      • x + y = 2
      • 2x + 2y = 4
      • This system has infinitely many solutions, expressed parametrically as x = t; y = 2 - t.

    Gaussian Elimination

    • This method is used for systems of three or more variables.

    • It involves transforming the system of equations into a triangular form using row operations to simplify the matrix of coefficients.

    • Example:

      • x – 2y + 3z = 9
      • -x + 3y = -4
      • 2x – 5y + z = 17
      • Solving the above equations by Gaussian Elimination yields x = 1, y = 2, z = 2

    Homework Assignment

    • Solve the following system of linear equations: x + 2y + 3z = 3, 2x + 5y + 7z = 6, 3x + 7y + 8z = 5.

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    Related Documents

    Lecture 5 Gauss Elimination PDF

    Description

    This quiz covers the solution of systems of linear equations, including methods such as Gaussian elimination and substitution. You will explore examples that illustrate systems with one solution, no solution, or infinitely many solutions. Test your understanding of these concepts through practical examples.

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