Systems of Equations with Infinite Solutions
SAT Big Idea: A system of linear equations has infinitely many solutions when both equations describe the exact same line. Every point on that line satisfies both equations simultaneously.
What Does "Infinite Solutions" Mean?
Consider a system of two linear equations:
$$\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$$
There are three possible outcomes:
| Outcome | Graphically | # of Solutions |
|---|---|---|
| Lines intersect | Cross at one point | Exactly 1 |
| Lines are parallel | Never touch | 0 (no solution) |
| Lines are identical | Overlap completely | Infinitely many |
When the system has infinite solutions, one equation is just a multiple of the other. They are the same line written differently.
Visual: Three Scenarios
Interactive visualization — available in the full lesson
Interactive visualization — available in the full lesson