16.2.1: Resources and Key Concepts

Math 375: Pre-Calculus · LibreTexts

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    From LibreTexts, under not declared, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    Key Concepts

    Definitions

    • Upper Triangular Form: A system of linear equations is in upper triangular form if the variables are eliminated from the equations in a way that creates a "triangle" shape. For a 3x3 system, the first equation may contain \(x, y, z\); the second contains only \(y, z\); and the third contains only \(z\).
    • Gaussian Elimination: An algorithmic process used to solve systems of linear equations by applying a series of row operations to transform the system into upper triangular form.
    • Ordered Triple: An ordered collection of three elements, \((x, y, z)\), that represents the solution to a system of three equations in three variables. Graphically, it is the point of intersection of three planes in space.

    Theorems

    • Invariant Solutions Under Row Operations: The solution to a system of linear equations remains unchanged (is invariant) under the following operations:
      1. Row Exchange: Interchange the order of any two equations.
      2. Row Scaling: Multiply both sides of an equation by a nonzero constant.
      3. Row Replacement: Add a nonzero multiple of one equation to another equation.
    • Number of Solutions to 3x3 Linear Systems: A system of three linear equations involving three unknowns has either one solution, no solution, or infinitely many solutions.

    Common Mistakes

    • Not Eliminating the Same Variable: When using elimination, a common error is to eliminate one variable from a first pair of equations (e.g., eliminate \(x\) from E1 and E2) but a different variable from a second pair (e.g., eliminate \(y\) from E1 and E3). This does not successfully reduce the system to a 2x2 system.
    • Arithmetic Errors in Row Operations: Making simple addition or multiplication errors when performing row scaling or row replacement. For example, when multiplying an entire equation by a constant, forgetting to multiply the term on the other side of the equals sign.
    • Sign Errors During Elimination: Forgetting to distribute a negative sign correctly when adding a multiple of one equation to another, leading to incorrect cancellation of terms.
    • Incorrectly Expressing Dependent Solutions: When a system is found to be dependent (infinite solutions), stating "infinite solutions" is not enough. A common mistake is not expressing the general solution as an ordered triple in terms of one of the variables (e.g., \((x, \frac{5}{2}x, \frac{3}{2}x)\)).
    • Misinterpreting Contradictions and Identities:
      • Arriving at a contradiction (e.g., \(0=2\)) and not concluding that the system is inconsistent and has no solution.
      • Arriving at an identity (e.g., \(0=0\)) and not concluding that the system is dependent and has infinite solutions.
    • Errors in Back-Substitution: After finding the value of the first variable (e.g., \(z\)) from the upper triangular form, making arithmetic errors when substituting it back into the other equations to find the remaining variables.

    This page titled 16.2.1: Resources and Key Concepts was last modified on Tue, 08 Jul 2025 17:48:22 GMT and is shared under a not declared license and was authored, remixed, and/or curated by Roy Simpson.