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3 Responses
You’re not messing up-those two expressions are the same. From P = 2l + 2w, one quick route is to divide the whole equation by 2 right away: P/2 = l + w, then subtract w to get l = P/2 − w. Your route is fine too: subtract 2w to get P − 2w = 2l, then divide by 2 to get l = (P − 2w)/2. If you “split” that fraction, (P − 2w)/2 = P/2 − (2w)/2 = P/2 − w. I always hesitate for a second here because P might not be a multiple of 2, but that’s okay-you just leave it as P/2. The only time this kind of splitting doesn’t behave is when things are multiplied together in the numerator (like (P·2w)/2), not when they’re added or subtracted. I sometimes even think it gets messy if there are three terms, and you can’t split cleanly-but that’s me overthinking it; the “split” works term-by-term for addition and subtraction.
Tiny analogy: imagine halving a bill. Half of “P dollars minus 2w dollars” is the same as “half of P dollars minus half of 2w dollars.” Same idea here. If you want a neat refresher on rearranging formulas, Khan Academy has a nice walkthrough: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:literal-equations/x2f8bb11595b61c86:rearrange-formulas/v/rearrange-formulas-example
You’ve got it right! Starting from P = 2l + 2w, subtract 2w to get P − 2w = 2l, then divide by 2 to get l = (P − 2w)/2. The key’s version l = P/2 − w is the same thing-just written after splitting the division across the subtraction: (P − 2w)/2 = P/2 − (2w)/2 = P/2 − w. Another quick path (because factoring is a tiny superpower) is to rewrite the right side as 2(l + w), so P = 2(l + w) ⇒ l + w = P/2 ⇒ l = P/2 − w. Same destination, two scenic routes.
Quick check with numbers: say P = 30 and w = 4. Your form: l = (30 − 8)/2 = 22/2 = 11. The key’s form: l = 30/2 − 4 = 15 − 4 = 11. Matching answers-high five! The only “gotcha” is to remember that dividing the whole side by 2 means every term on that side gets divided, unless you’ve already factored that 2 out. Parentheses keep everything tidy.
Yep-those are the same: (P − 2w)/2 = P/2 − (2w)/2 = P/2 − w, so you’re just distributing the division across the sum (like a reverse distributive move). For example, if P=20 and w=3, then (20−6)/2=14/2=7 and also 20/2−3=10−3=7.