Question:16x - 3 = 4x + 6Which equation has the same solution as the given equation?

1. INFER what the question is asking

• The question asks for an equation with the "same solution" as the given equation
• This means we need an equivalent equation - one that yields the identical x-value when solved
• Key insight: We can create equivalent equations by applying valid algebraic operations to both sides

2. SIMPLIFY the original equation using algebraic properties

• Given: \(16\mathrm{x} - 3 = 4\mathrm{x} + 6\)
• Subtract \(4\mathrm{x}\) from both sides to collect all variable terms on the left:
  \(16\mathrm{x} - 4\mathrm{x} - 3 = 4\mathrm{x} - 4\mathrm{x} + 6\)
  \(12\mathrm{x} - 3 = 6\)
• Add 3 to both sides to isolate the variable term:
  \(12\mathrm{x} - 3 + 3 = 6 + 3\)
  \(12\mathrm{x} = 9\)

3. INFER the answer from our work

• We've transformed the original equation into \(12\mathrm{x} = 9\) through valid algebraic steps
• This matches answer choice (A) exactly
• Since we used valid operations, this equation must have the same solution as the original

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not understanding that "same solution" means equivalent equations, so they try to solve each answer choice individually instead of transforming the original equation.

Students might solve the original equation to get \(\mathrm{x} = \frac{3}{4}\), then test each answer choice by substitution. While this works, it's inefficient and prone to calculation errors. More critically, they miss the conceptual point that algebraic transformations create equivalent equations.

This approach could lead to computational mistakes that cause them to select Choice B (\(12\mathrm{x} = 6\)) or other incorrect options.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors during the algebraic manipulation steps.

Common mistakes include calculating \(16\mathrm{x} - 4\mathrm{x} = 11\mathrm{x}\) instead of \(12\mathrm{x}\), or incorrectly handling the constants when adding/subtracting. These computational errors lead directly to wrong simplified equations.

This may lead them to select Choice C (\(20\mathrm{x} = 9\)) or Choice D (\(20\mathrm{x} = 6\)) depending on the specific error made.

The Bottom Line:

This problem tests both conceptual understanding of equivalent equations and technical skill in algebraic manipulation. Success requires recognizing that the most efficient approach is transforming the original equation, not testing each answer choice separately.