A farmer has 100 m of fence for a rectangular pen against a river (no fence needed on the river side). Order the optimization workflow.

1. Name variables: width $x$, length $y$, area $A = xy$
2. Encode the constraint: $2x + y = 100$
3. Substitute to get one variable: $A(x) = x(100 - 2x)$
4. Set $A'(x) = 100 - 4x = 0$, giving $x = 25$
5. Confirm it is a maximum and report $25 \times 50$ m

Every optimization problem walks these same five steps; only the story changes. Note the punchline $y = 2x$: the free river side lets the pen stretch twice as long as it is deep.