A flowchart represents the equation: x → ×4 → +7 → result is 27. To solve for x, in what order should the student reverse the operations?
AFirst divide by 4, then subtract 7
BFirst subtract 7, then divide by 4
CFirst add 7, then multiply by 4
DFirst multiply by 4, then subtract 7
The flowchart shows that multiplication was applied first, then addition. To reverse it, undo the operations in reverse order: subtract 7 first (the last thing done), then divide by 4 (the first thing done). 27 − 7 = 20; 20 ÷ 4 = 5. Dividing first (option A) is the most common error — students see the coefficient and want to deal with it immediately, but it's actually the inner wrapping that comes off last.
Question 2 Multiple Choice
A student solves 3x + 6 = 21 by first dividing both sides by 3 to get x + 2 = 7, then subtracting 2 to get x = 5. A second student subtracts 6 first to get 3x = 15, then divides by 3 to get x = 5. Which of the following is true?
AOnly the second student used a valid strategy
BOnly the first student used a valid strategy
CBoth strategies are algebraically valid and produce the correct answer; the second is the standard approach
DNeither strategy is correct because the distributive property must be applied first
Both strategies are algebraically valid — the balance principle applies regardless of which operation you undo first. However, dividing through first can create fractions when the coefficient doesn't divide evenly into the constant term (e.g., 3x + 7 = 20 becomes x + 7/3 = 20/3 if you divide first). Undoing addition/subtraction first is the standard approach precisely because it keeps the numbers cleaner and applies consistently across all cases.
Question 3 True / False
When solving a two-step equation, you should undo the operation that was applied LAST when the equation was constructed.
TTrue
FFalse
Answer: True
This is the core principle. If the equation was built by first multiplying x, then adding a constant, the addition is the 'outer layer' — the last thing done and the first thing undone. The flowchart model makes this concrete: run the arrows backwards, reversing each operation in reverse sequence.
Question 4 True / False
In the equation 4x − 9 = 11, the first step should be to divide both sides by 4, because the coefficient is the most prominent operation in the expression 4x.
TTrue
FFalse
Answer: False
This reverses the correct order. The standard first step is to add 9 to both sides (4x = 20), then divide by 4 (x = 5). Dividing first gives x − 9/4 = 11/4, which introduces fractions unnecessarily. The subtraction of 9 is the 'outermost wrapping' — it was applied last and should be undone first.
Question 5 Short Answer
Explain why solving two-step equations requires working in 'reverse order of operations.' What does 'reverse' mean here, and what mistake does ignoring this principle typically cause?
Think about your answer, then reveal below.
Model answer: An equation like 2x + 3 = 11 was built by starting with x, multiplying by 2, then adding 3. Reversing means undoing those steps in the opposite sequence: first subtract 3 (undo the addition), then divide by 2 (undo the multiplication). Ignoring this — dividing by 2 first — either produces a messier equation with fractions or, if the student makes an error, gives the wrong answer. The reverse-order principle ensures you always remove the outermost layer first, just like unwrapping a package from the outside in.
The 'reverse order' mirrors how inverse functions work in later mathematics. The same logic applies in multi-step equations, function composition, and even calculus (the chain rule). Building the habit now — always ask 'what was done last?' before picking your first move — prevents the most common procedural error in equation solving.