y - 57 = px The given equation relates the positive numbers p, x, and y. Which equation correctly expresses...

1. INFER the goal

• Given: \(\mathrm{y - 57 = px}\)
• Need to: Express y in terms of p and x
• What this means: Get y by itself on one side of the equation

2. INFER the strategy

• Since we have \(\mathrm{y - 57 = px}\), we need to eliminate the "-57" from the left side
• Strategy: Add 57 to both sides to isolate y

3. SIMPLIFY through algebraic manipulation

• Add 57 to both sides:
  \(\mathrm{y - 57 + 57 = px + 57}\)
• Left side simplifies: \(\mathrm{y - 57 + 57 = y}\)
• Right side stays: \(\mathrm{px + 57}\)
• Result: \(\mathrm{y = px + 57}\)

Answer: B. \(\mathrm{y = px + 57}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't clearly understand what "express y in terms of p and x" means, leading them to attempt unnecessary manipulations or get confused about the goal.

Some students might think they need to rearrange the terms differently, potentially leading them to select Choice A (\(\mathrm{y = 57x + p}\)) by incorrectly switching the roles of the variables and constants.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors during the algebraic manipulation, such as subtracting 57 instead of adding it, or incorrectly handling the variable terms.

This algebraic confusion might lead them to select Choice C (\(\mathrm{y = 57px}\)) if they incorrectly think they need to multiply the terms together, or Choice D (\(\mathrm{y = px/57}\)) if they mistakenly believe division is required.

The Bottom Line:

This problem tests whether students understand the fundamental goal of "solving for" a variable and can execute basic algebraic steps accurately. The key insight is recognizing that isolating y requires undoing the subtraction of 57.