If 6 + x = 9, what is the value of 18 + 3x?

1. TRANSLATE the problem information

• Given equation: \(6 + \mathrm{x} = 9\)
• Find: The value of \(18 + 3\mathrm{x}\)

2. INFER the most efficient approach

• Notice that \(18 + 3\mathrm{x}\) looks similar to \(3(6 + \mathrm{x})\)
• Instead of solving for x first, we can multiply the entire equation by 3
• This will directly give us the expression we want

3. SIMPLIFY by applying equation manipulation

• Multiply both sides of \(6 + \mathrm{x} = 9\) by 3:
  \(3(6 + \mathrm{x}) = 3(9)\)

• Apply the distributive property on the left side:
  \(18 + 3\mathrm{x} = 27\)

Answer: 27

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the efficient strategy and instead solve for x first.

They solve \(6 + \mathrm{x} = 9\) to get \(\mathrm{x} = 3\), then calculate \(18 + 3\mathrm{x} = 18 + 3(3) = 27\). While this gives the correct answer, it takes more steps and misses the elegant algebraic insight. This approach works but demonstrates less sophisticated algebraic thinking.

Second Most Common Error:

Poor SIMPLIFY execution: Students attempt the correct strategy but make arithmetic errors.

They correctly set up \(3(6 + \mathrm{x}) = 3(9)\) but make mistakes in the distributive property (perhaps getting \(18 + \mathrm{x} = 27\)) or in the final arithmetic. This leads to confusion and potentially incorrect answers.

The Bottom Line:

This problem rewards students who can see algebraic patterns and manipulate equations strategically rather than just mechanically solving step-by-step. The key insight is recognizing that the target expression \(18 + 3\mathrm{x}\) is exactly what you get when you multiply the given equation by 3.