Perimeter as an Expression

Sometimes a side is not a number but an expression. A rectangle is w wide, and its length is 5 more than its width, so w + 5. Its perimeter is two widths plus two lengths: $2w+2\left(w+5\right)$.

This simplifies with the tools from the earlier chapters: distribute, then collect like terms. $2\left(w+5\right)$ is $2w+10$, so the perimeter is $2w+2w+10$, and the two w terms make $4w+10$.

The short form works for every rectangle of this shape. A width of 3 gives $4\cdot 3+10$, which is 22. Why simplify? The short form takes one multiplication and one addition, and it shows at a glance how the perimeter grows: by 4 for every extra unit of width.