Question:4y = 108What value of y is the solution to the given equation?Enter your answer as an integer.

1. INFER the solving strategy

• Given: \(\mathrm{4y = 108}\)
• Goal: Find the value of y
• Strategy: Since y is multiplied by 4, I need to "undo" this multiplication by dividing both sides by 4

2. SIMPLIFY by applying the division

• Divide both sides of the equation by 4:
  \(\mathrm{4y ÷ 4 = 108 ÷ 4}\)
• This gives us: \(\mathrm{y = 27}\)

3. INFER the need to verify

• Substitute \(\mathrm{y = 27}\) back into the original equation:
  \(\mathrm{4(27) = 108}\) ✓
• Since this equals 108, our answer is correct

Answer: 27

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that division is needed to isolate y. Instead, they might try subtracting 4 from both sides, thinking: "I need to get rid of the 4, so I'll subtract it."

This leads them to: \(\mathrm{4y - 4 = 108 - 4}\), giving \(\mathrm{4y = 104}\), then getting stuck or continuing with incorrect operations. This causes confusion and leads to guessing or selecting an incorrect answer like 104 or 26.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that they need to divide by 4, but make an arithmetic error when calculating \(\mathrm{108 ÷ 4}\). They might rush through the division and get 26 instead of 27, or make other calculation mistakes.

This leads them to an incorrect final answer that seems reasonable but fails verification.

The Bottom Line:

This problem tests whether students understand that solving equations requires undoing operations in reverse order, and whether they can execute basic algebraic manipulations accurately. The key insight is recognizing that multiplication by 4 requires division by 4 to isolate the variable.