If 2x + 3 = 9, what is the value of 6x - 1?

1. TRANSLATE the problem information

• Given: \(\mathrm{2x + 3 = 9}\)
• Find: The value of \(\mathrm{6x - 1}\)

2. INFER the best approach

You have two solid options here:

• Option A: Solve for x first, then substitute into \(\mathrm{6x - 1}\)
• Option B: Manipulate the given equation directly to create \(\mathrm{6x - 1}\)

Let's explore both approaches:

3. SIMPLIFY using Option A (Solve for x first)

From \(\mathrm{2x + 3 = 9}\):

• Subtract 3 from both sides: \(\mathrm{2x = 6}\)
• Divide both sides by 2: \(\mathrm{x = 3}\)
• Substitute into target expression:
  \(\mathrm{6x - 1 = 6(3) - 1}\)
  \(\mathrm{= 18 - 1}\)
  \(\mathrm{= 17}\)

4. SIMPLIFY using Option B (Direct manipulation)

Starting with \(\mathrm{2x + 3 = 9}\):

• Multiply both sides by 3: \(\mathrm{3(2x + 3) = 3(9)}\)
• This gives us: \(\mathrm{6x + 9 = 27}\)
• Subtract 10 from both sides: \(\mathrm{6x + 9 - 10 = 27 - 10}\)
• Final result: \(\mathrm{6x - 1 = 17}\)

Answer: 17

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when solving \(\mathrm{2x + 3 = 9}\)

They might incorrectly get \(\mathrm{x = 2}\) (forgetting to subtract 3 first) or \(\mathrm{x = 4.5}\) (adding instead of subtracting). When they substitute these wrong x-values into \(\mathrm{6x - 1}\), they get answers like 11 or 26, leading to confusion since these aren't typical answer choices.

Second Most Common Error:

Limited INFER reasoning: Students don't recognize the direct manipulation approach exists

They only see the "solve for x first" method and may get overwhelmed by the multi-step process. Some students abandon the systematic approach entirely when they realize there are multiple steps involved, leading to random guessing.

The Bottom Line:

This problem rewards students who can either execute multi-step algebra cleanly OR recognize that clever equation manipulation can bypass finding x explicitly. The key insight is that both paths are valid - success comes from choosing one and executing it carefully.