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System of Equations Geometry

Run the System of Equations Geometry MicroSim Fullscreen

Edit the MicroSim with the p5.js editor

About This MicroSim

This interactive visualization helps students understand how the solution to a system of linear equations corresponds to the geometric intersection of lines on a coordinate plane.

Key Features:

  • Line Visualization: See two linear equations displayed as colored lines on a coordinate grid
  • Real-time Updates: Adjust coefficients with sliders and watch the geometry change instantly
  • Solution Detection: Automatically identifies unique solutions, infinite solutions (coincident lines), or no solution (parallel lines)
  • Random Generator: Create random systems that always have solutions within the visible grid

How to Use

  1. Adjust Coefficients: Use the sliders to change the values of a, b, and c for each equation in the form ax + by = c
  2. View Solution: The green point marks the intersection (solution) when one exists
  3. Generate Random: Click "Random" to create a new system with a guaranteed solution
  4. Explore Cases: Try creating parallel lines (no solution) or coincident lines (infinite solutions)

Solution Types

Configuration Geometric Interpretation Algebraic Meaning
Lines intersect at one point Unique solution det(A) ≠ 0
Lines are parallel No solution det(A) = 0, inconsistent
Lines are coincident Infinite solutions det(A) = 0, consistent

Lesson Plan

Learning Objectives

After using this MicroSim, students will be able to:

  1. Interpret the solution of a system of linear equations geometrically
  2. Identify when systems have unique, infinite, or no solutions
  3. Connect the algebraic determinant condition to geometric configuration

Suggested Activities

  1. Exploration: Start with the default system (x + y = 3, x - y = 1) and verify the solution (2, 1) by substitution
  2. Create Parallel Lines: Adjust coefficients to make the lines parallel. What pattern do you notice in the coefficients?
  3. Create Coincident Lines: Make both equations represent the same line. How are the coefficients related?
  4. Predict Before Moving: Before adjusting a slider, predict how the line will change

Discussion Questions

  • How does changing the constant term c affect the line's position?
  • What coefficient patterns lead to parallel lines?
  • Why does the determinant being zero correspond to parallel or coincident lines?