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1: Solving Systems of Linear Equations

  • Page ID
    201831
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    • 1.1: Systems of Linear Equations
      This page discusses linear equations and systems, defining them and their classifications based on the existence and uniqueness of solutions. It covers geometric interpretations for two-variable systems (intersecting, parallel, or coinciding lines) and extends to three-variable systems (intersecting, non-intersecting, or coinciding planes). The text emphasizes the complexity that arises with additional variables, necessitating linear algebra for further analysis.
    • 1.2: Matrix notation and Row Reduction
      This page presents a comprehensive overview of solving systems of linear equations using augmented matrices and row operations. Key techniques include elementary row operations, achieving row echelon form (REF) and reduced row echelon form (RREF), and understanding the implications of basic and free variables. The document includes examples that demonstrate the application of these techniques, identifies cases of infinite solutions via parameters, and addresses practical mathematical problems.
    • 1.3: Rank and Homogeneous Systems
      This page discusses homogeneous systems of linear equations, emphasizing the presence of the trivial solution and the conditions for nontrivial solutions when there are more variables than equations. It explains the rank of a matrix as a measure of linear independence, its derivation from pivot columns, and its importance in determining the nature of solutions.
    • 1.4: Chemistry Application- Balancing Chemical Reactions
      The tools of linear algebra can also be used in the subject area of Chemistry, specifically for balancing chemical reactions.
    • 1.5: Applications to Physics and Resistor Networks
      The tools of linear algebra can be used to study the application of resistor networks.
    • 1.6: Chapter Exercises

    Thumbnail: A linear system in three variables determines a collection of planes. The intersection point is the solution. (CC BY-SA 4.0; Fred the Oyster via Wikipedia)


    This page titled 1: Solving Systems of Linear Equations is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Ken Kuttler (Lyryx) via source content that was edited to the style and standards of the LibreTexts platform.

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