Drawing Conclusions from Graphs in the Coordinate Plane (AF 603, AF 704)

A graph of two lines tells two separate stories at once. Each line is its own rule: an output for every input. Neither line alone can tell you when the two stories agree, because each only describes itself. The crossing point is the single input where both outputs match. Before that input one line sits higher, after it the other does, and the crossing is the moment the lead changes hands.

To find that moment without drawing anything, write both rules as expressions in the same variable and set them equal. Solving that equation gives the input where they meet; plug it back into either rule to get the shared output. This works because a graph is just a picture of an equation, so a shared point on both graphs is a shared solution to both equations at once.

This sits right next to reading slope and intercept as a story: the flat fee is where a line starts, the rate is how fast it climbs, and the intersection is where two starting points and two climb rates finally balance out. Raising a flat fee shifts a line straight up without changing its steepness, which moves the crossing point but not the basic idea that steeper-but-cheaper-per-unit rules eventually catch up to flatter-but-pricier ones.