Dynamic Exploration of Linear Systems and Coefficient–Solution Links

This GeoGebra example dynamically visualizes how changing coefficients affects the solutions of a system of linear equations by showing the interaction between two lines—intersection, parallelism, or coincidence—and the corresponding algebraic solution behavior. Key math concepts include systems of linear equations, linear (two-variable) equations, line graphs in the coordinate plane, intersection-point interpretation, and the relationship between coefficients and the number/location of solutions, with proficiency checked via both the interactive app and algebraic techniques. Construction and interactivity are driven by commands and objects from the prototype: Shuffle/Sequence to generate or iterate cases (allCaseNum, caseNum, countQuestion), Element and SelectedIndex to select the active problem instance (ddMenu, btnNew/textQuestion/textAllIn), Polygon/Segment/Intersect and PointIn to build geometric entities and determine intersection/contact conditions, Polynomial to represent the analytic form of relationships, Join to link related objects, and UI/visual controls like sliders and animation objects (aSli/bSli/cSli, btnAnim, aniSli, scriptResetApp, pulsing) along with coordinate/frame settings (xMin/xMax, yMin/yMax, xGrid/yGrid, GRMIN/GRMAX, labelX/labelY). It is intended for middle and high school through early undergraduate learners for classroom demonstrations and structured practice.