Operation Order Reversal

Pre-Algebra · Axiom Academy

LESSON Undoing in Reverse Order To isolate a variable, undo each operation in the opposite order it was applied — peel off the last one first. 1. How the Expression Was Built Watch an input flow through two operations in PEMDAS order : first multiply by 2, then add 3. Tracking a sample value x = 5 alongside the symbols, the chain produces 2x + 3 , which lands on 13 . Step 1 (×) happens before Step 2 (+) Now solve 2x + 3 = 13 for x . The build order was × then + ; to take it apart we run that backwards — undo the + first, then undo the × . Subtract 3 from both sides ( ), then divide both sides by 2 ( ). The last operation applied is the first one undone. 3. Why the Order Can't Be Swapped What if we got impatient and divided by 2 before subtracting 3? The two undo-paths from 13 race side by side. The correct order ( -3 , then ) lands on 5 . The swapped order ( , then -3 ) overshoots and lands on 3.5 . The reverse-order rule scales to any number of operations. Here x is built by three: multiply by 3, add 4, then divide by 2, giving . To solve , undo them in reverse : multiply by 2, subtract 4, divide by 3. Built ×3 → +4 → ÷2, so undo ÷2 → −4 → ÷3 Multiply both sides by 2: the outermost operation comes off first. Subtract 4 from both sides, peeling the next layer. Divide both sides by 3 — the first operation applied is the last one undone. Three operations or thirty: always strip the most-recent one off first.

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