Number of Solutions of a Linear System of Two Equations (Geometric Test)

This example demonstrates, on the same coordinate plane, how the relative position of two lines determines the number of solutions of a simultaneous linear equation system: exactly one solution, no solution, or infinitely many solutions. The key mathematical ideas are the geometric meaning of solving linear equations, the correspondence between intersection points of two lines and solution counts, and how varying parameters changes the lines and thus the intersection behavior. Construction highlights include using GeoGebra commands such as Sequence to generate parameter-related sequences, Textfield together with InputBox1 and InputBox2 for user-controlled coefficients/conditions and dynamic labels, Line and Corner to build the geometric configuration and key reference points/objects (e.g., chi, g, f, A_0, A_1, t_1, t_2, etc.), and using multiple constructed objects like F_1–F_3 and G_1–G_3 along with point coordinates (x_1…x_6, y_1, y_2) to realize the two-line system and observe how the number of intersections changes. Suitable for high school through early college, especially for teaching the geometric interpretation and solution-count criteria of 2×2 linear systems.