If \(5 - 7(2 - 4\mathrm{x}) = 16 - 8(2 - 4\mathrm{x})\), what is the value of 2 - 4x?

1. TRANSLATE the problem information

• Given equation: \(5 - 7(2 - 4\mathrm{x}) = 16 - 8(2 - 4\mathrm{x})\)
• Find: the value of \((2 - 4\mathrm{x})\)

2. INFER the key insight

• Notice that the expression \((2 - 4\mathrm{x})\) appears on both sides of the equation
• This means we can treat \((2 - 4\mathrm{x})\) as a single unit, like substituting a variable
• Let's call this expression "u" where \(\mathrm{u} = 2 - 4\mathrm{x}\)

3. SIMPLIFY by substitution

• Our equation becomes: \(5 - 7\mathrm{u} = 16 - 8\mathrm{u}\)
• Now we can solve for u using standard algebraic steps

4. SIMPLIFY the equation

• Starting with: \(5 - 7\mathrm{u} = 16 - 8\mathrm{u}\)
• Add 8u to both sides: \(5 - 7\mathrm{u} + 8\mathrm{u} = 16 - 8\mathrm{u} + 8\mathrm{u}\)
• This gives us: \(5 + \mathrm{u} = 16\)
• Subtract 5 from both sides: \(\mathrm{u} = 11\)

5. INFER the final answer

• Since \(\mathrm{u} = 2 - 4\mathrm{x}\), we have \(2 - 4\mathrm{x} = 11\)

Answer: 11

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \((2 - 4\mathrm{x})\) can be treated as a single unit and instead try to distribute and expand everything immediately.

They might expand: \(5 - 14 + 28\mathrm{x} = 16 - 16 + 32\mathrm{x}\), getting \(-9 + 28\mathrm{x} = 32\mathrm{x}\), then \(-9 = 4\mathrm{x}\), leading to \(\mathrm{x} = -9/4\). But the question asks for \(2 - 4\mathrm{x}\), not x, so they'd substitute back incorrectly or forget this step entirely. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students recognize the substitution approach but make sign errors during algebraic manipulation.

Common mistake: When moving terms, they might write \(5 - 7\mathrm{u} = 16 - 8\mathrm{u}\), then incorrectly get \(5 + 8\mathrm{u} = 16 + 7\mathrm{u}\), leading to \(\mathrm{u} = 11/15\) instead of \(\mathrm{u} = 11\). This causes them to get stuck and guess.

The Bottom Line:

The key insight is recognizing patterns in equations - when the same expression appears multiple times, treat it as a single unit rather than expanding everything. This dramatically simplifies what could otherwise become a messy algebraic manipulation.