If 2x - 1 = 5, what is the value of 6x - 3?

1. INFER the solution strategy

• Given: \(\mathrm{2x - 1 = 5}\)
• Need to find: \(\mathrm{6x - 3}\)
• Key insight: We must solve for \(\mathrm{x}\) first, then substitute that value into the target expression

2. SIMPLIFY the given equation to find x

• Start with: \(\mathrm{2x - 1 = 5}\)
• Add 1 to both sides: \(\mathrm{2x = 6}\)
• Divide both sides by 2: \(\mathrm{x = 3}\)

3. SIMPLIFY by substituting x = 3 into the target expression

• \(\mathrm{6x - 3 = 6(3) - 3}\)
  \(\mathrm{= 18 - 3}\)
  \(\mathrm{= 15}\)

Answer: 15

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to work directly with \(\mathrm{6x - 3}\) without first solving for \(\mathrm{x}\)

Students might attempt to manipulate \(\mathrm{6x - 3}\) in isolation, perhaps trying to relate it to \(\mathrm{2x - 1}\) without a clear strategy. They might write something like "\(\mathrm{6x - 3}\) is three times \(\mathrm{2x - 1}\)" but then get confused about how to use this relationship systematically.

This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic mistakes when solving for \(\mathrm{x}\) or substituting

Students correctly identify they need to solve for \(\mathrm{x}\) first, but then make calculation errors like:

• \(\mathrm{2x - 1 = 5 → 2x = 4}\) (forgetting to add 1 correctly)
• Or correctly finding \(\mathrm{x = 3}\) but then computing \(\mathrm{6(3) - 3 = 21}\) (incorrectly calculating \(\mathrm{18 - 3}\))

This leads to selecting an incorrect numerical answer based on their computational errors.

The Bottom Line:

This problem tests whether students can connect a solved equation to a related expression requiring the same variable. The key insight is recognizing that finding \(\mathrm{6x - 3}\) depends on first determining what \(\mathrm{x}\) equals from the given equation.