Analysis of student performance across typical Algebra 1 cohorts reveals clear patterns in where Homework 5 breaks down. The errors are rarely conceptual failures about what equality means; they are execution slips occurring early in the algebraic sequence.
| Problem Structure | Frequent Student Pitfall | Correct Mechanical Execution |
|---|---|---|
| $-3(2x - 5) = 27$(Negative Distribution) | Multiplying $-3 \cdot (-5)$ and writing $-15$ instead of $+15$. | Distribute $-3$ to both terms: $-6x + 15 = 27$, leading to $x = -2$. |
| $5x - 8 + 2x = 3x + 12$(Combining Across Equals Sign) | Subtracting $2x$ from $5x$ on the same side as if crossing the equals boundary. | Group identical sides first: $7x - 8 = 3x + 12$, then subtract $3x$ across the sign. |
| $2(3x + 4) = 6x + 8$(Special Solutions: Identities) | Attempting to solve for $x$ and writing $x = 0$ or becoming confused by $8 = 8$. | Recognize an identity where both expressions are identical: Infinite Solutions ($\mathbb{R}$). |
| $4x - 7 = 4x + 3$(Special Solutions: Contradictions) | Subtracting $4x$ and writing $x = -10$ or assuming an algebraic mistake occurred. | Recognize a false statement ($-7 = 3$) upon canceling variables: No Solution ($\emptyset$). |
