p + 34 = q + rThe given equation relates the variables p, q, and r. Which equation correctly expresses...

1. INFER the approach needed

• Given: \(\mathrm{p + 34 = q + r}\)
• Goal: Express p in terms of q and r
• Strategy: Since 34 is added to p, use the inverse operation (subtraction) to isolate p

2. SIMPLIFY by applying inverse operations

• Subtract 34 from both sides of the equation:
  \(\mathrm{p + 34 - 34 = q + r - 34}\)
• Simplify the left side:
  \(\mathrm{p = q + r - 34}\)

Answer: B. p = q + r - 34

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students recognize they need to "get rid of" the 34, but incorrectly think they should add 34 to both sides instead of subtract.

Their thinking: "I see +34 with p, so I'll add 34 to the other side too."
This gives them: \(\mathrm{p + 34 + 34 = q + r + 34}\), which leads to \(\mathrm{p = q + r + 34}\)

This may lead them to select Choice A (p = q + r + 34)

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify they need to subtract 34, but make sign errors when manipulating the equation, incorrectly distributing negative signs to q and r.

Their thinking: "Subtract 34 from both sides" but then write \(\mathrm{p = -(q + r) + 34}\) or similar variations.

This may lead them to select Choice C (p = -q - r + 34) or Choice D (p = -q - r - 34)

The Bottom Line:

This problem tests whether students understand inverse operations for isolating variables. The key insight is that to undo addition, you subtract - and you must do this to both sides of the equation to maintain equality.