1. Build a system
The system is
\(a_1x+b_1y=c_1\)
\(a_2x+b_2y=c_2\).
Equation 1
a₁
b₁
c₁
Equation 2
a₂
b₂
c₂
2. Geometry
Blue and orange lines represent the two equations. The black point appears when there is a unique solution.
3. Rank test and RREF
The coefficient matrix and augmented matrix are
For a system \(Ax=b\), the system is consistent exactly when \(\operatorname{rank}(A)=\operatorname{rank}([A\mid b])\). For two variables, a unique solution occurs when this common rank is \(2\).
4. Student practice tasks
Task A. Use the sliders to create a system with exactly one solution.
Record \(A\), \(b\), \(\operatorname{rank}(A)\), \(\operatorname{rank}([A\mid b])\), and the solution.
Task B. Use the sliders to create a system with no solution.
Explain the geometry and the rank test.
Task C. Use the sliders to create a system with infinitely many solutions.
Explain why the two equations define the same line.
Task D. Change only $c_2$. What changes geometrically? What changes algebraically?
5. Short-answer workspace
This space is for practice only. Students should copy their final answers into the course submission form or notebook.