About Course

This course provides a rigorous introduction to the theory and methods for solving systems of equations. It begins with a formal definition of a system of equations and classification into consistent, inconsistent, and dependent systems. Students will master five distinct analytic techniques for solving 2×2 linear systems—graphical interpretation, substitution, elimination, Cramer’s rule via determinants, and Gauss–Jordan elimination—highlighting the prerequisites and scope of each method. The curriculum extends these methods to 3×3 systems, with particular attention to determinant-based solutions and row-reduction algorithms. Finally, learners will confront nonlinear systems involving one or two quadratic equations, employing algebraic techniques to characterize and compute their solutions. 29.99

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What Will You Learn?

  • Precisely define and classify linear systems and their matrix representations
  • Solve 2×2 systems graphically and confirm solutions algebraically
  • Apply substitution and elimination to both 2×2 and 3×3 systems
  • Use Cramer’s rule to compute solutions via determinants, understanding its domain of validity
  • Perform Gauss–Jordan elimination to reduce any linear system to reduced row-echelon form
  • Extend these techniques to solve systems involving quadratic equations
  • Select the most efficient method based on system size and structure
  • Verify and interpret solutions in both mathematical and applied contexts

Course Content

1. Foundations of Systems of Equations
This topic establishes the formal definition of a system of linear equations, including notation for coefficient matrices, variable vectors, and constant terms. It also introduces the classification of systems as consistent, inconsistent, or dependent, and explains how these classifications relate to the properties of the associated matrix equation 𝐴X=𝑏.

  • 1.1 Definitions and Notation of Linear Equations
  • 1.2 Classification: Consistent vs. Inconsistent; Independent vs. Dependent
  • 1.3 Matrix Representation and the Equation 𝐴𝑥=𝑏

2. Solving 2×2 Linear Systems
Students will learn to interpret a 2×2 system geometrically as the intersection of two lines in the plane, appreciating when graphical solutions suffice and when algebraic methods are required. Five algebraic techniques—substitution, elimination, Cramer’s rule, and Gauss–Jordan elimination—will be developed in detail, with attention to each method’s prerequisites, computational efficiency, and limitations.

3. Solving 3×3 Linear Systems
The methods for 2×2 systems are extended to three variables, emphasizing systematic approaches for hand calculation and laying groundwork for software implementation. Substitution and elimination techniques are presented alongside determinant-based Cramer’s rule and the algorithmic Gauss–Jordan reduction, with discussion of computational complexity and pivot strategies.

4. Word Problems for 2×2 and 3×3 Models
This topic focuses on translating diverse real-world scenarios into linear systems and solving them using the previously learned methods. Learners will model and solve 2×2 problems—mixture, age, and work-rate contexts—and extend their skills to 3×3 scenarios such as three-part mixtures, three-agent workforce problems, and capital allocation, emphasizing equation setup, solution techniques, and interpretation of results.

5. Nonlinear Systems of Equations
Students will apply linear techniques to systems in which one or more equations are quadratic, analyzing intersections of lines and conic sections. The topic concludes with challenge problems involving two quadratic equations, illustrating methods to reduce to univariate polynomial equations and interpret multiple real or complex solutions.

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