Systems of Non-Linear Equations

Introduction

  • The session starts with a brief greeting and check on students' well-being.

  • Activity is introduced to reinforce understanding of previous topics before moving on.

Activity

  • Participants instructed to join a quiz:

    • Website: joinmyquiz.com

    • Code: 330864

  • Duration of the activity is 10 minutes.

  • Objective: To review concepts of hyperbola learned previously through engaging activity.

Transition to New Topic

  • New Topic Introduced: System of Nonlinear Equations.

  • Connection with Previous Learning:

    • Explanation of nonlinear graphs (not straight lines, featuring curves or bends) in contrast to linear graphs (straight lines).

Understanding of Systems

  • Definition of Linear Systems:

    • A system of linear equations has equations of degree one.

    • Typically involves two or more equations.

  • Goals for Today's Lesson:

    • Identify nonlinear systems.

    • Learn to solve using substitution or elimination methods.

    • Solve problems involving nonlinear systems.

Graphical Interpretation of Systems

  • Types of Solutions for Linear Equations:

    1. No solution: Lines are parallel, never intersecting.

    2. One solution: Lines intersect at a single point.

    3. Infinite solutions: Lines coincide, overlap completely.

Explanation of Nonlinear Systems

  • Definition: A system is nonlinear if any of its equations are nonlinear.

  • Examples Provided:

    • Quadratic equations forming parabolas and their intersection with linear equations.

Methods for Solving Nonlinear Equations

  • 1. Graphing:

    • Identify the type of graph each equation forms before sketching.

    • Analyze intersection points to identify solutions.

  • 2. Elimination:

    • Method involves manipulating equations to eliminate one variable, solving for the other.

    • Ensure the equations are arranged to facilitate elimination.

  • 3. Substitution:

    • Solve one equation for one variable and substitute into the other equation.

Example Problems Discussed

  • Working through example systems, students are asked to describe each equation's graph and hypothesize about intersections.

  • Graph Outcomes:

    • Expect diagrams showcasing points of intersection.

    • Verification of solutions through substitution back into original equations.

Application Problems

  • Students engage with real-life examples, such as determining dimensions of a rectangular area given area and diagonal constraints.

  • Use substitution method to derive width and length from given conditions.

Concluding Remarks

  • Three Key Takeaways:

    • Methods to solve nonlinear equations include graphing, elimination, and substitution.

    • Practice is encouraged through assigned exercises that will be reviewed next class.

  • End of session with check for understanding through student reactions (thumbs up).

  • Reminder for students to work in pairs on exercises before the next class.