AMC 10 · 2003 · #9

The function is not handed over — only one fact about it is. Tool #4 (Introduce a Variable) writes the unknown line as f(x)=mx+b with both constants named, which makes the key event visible: b cancels out of every difference f(q)-f(p), so the missing information was never needed. Tool #5 (Look for a Pattern) supplies the reason behind the cancellation — on a straight line, equal runs always produce equal rises — and gives a second, formula-free route to the same number. Tool #3 (Eliminate Possibilities) is used at the end to see which wrong choices the standard slips would produce, since each distractor here matches a specific mis-scaling.