Which of the following is equivalent to \((4\mathrm{y} - 7) - (\mathrm{y} + 5)\)?3y - 163y - 123y - 23y...

1. TRANSLATE the problem information

• Given expression: \((4\mathrm{y} - 7) - (\mathrm{y} + 5)\)
• Goal: Simplify this to match one of the answer choices
• What this tells us: We need to remove parentheses and combine like terms

2. SIMPLIFY by distributing the subtraction

• When we subtract a quantity in parentheses, we subtract each term inside
• \((4\mathrm{y} - 7) - (\mathrm{y} + 5)\) becomes: \(4\mathrm{y} - 7 - \mathrm{y} - 5\)
• Key insight: The subtraction sign changes \(+5\) to \(-5\)

3. SIMPLIFY by combining like terms

• Group the variable terms: \(4\mathrm{y} - \mathrm{y} = 3\mathrm{y}\)
• Group the constant terms: \(-7 - 5 = -12\)
• Final result: \(3\mathrm{y} - 12\)

Answer: (B) \(3\mathrm{y} - 12\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly distribute the subtraction sign and write: \(4\mathrm{y} - 7 - \mathrm{y} + 5\)

Instead of changing \(+5\) to \(-5\), they keep it positive. This gives them:
\(4\mathrm{y} - \mathrm{y} + (-7 + 5) = 3\mathrm{y} - 2\)

This leads them to select Choice (C) \(3\mathrm{y} - 2\)

Second Most Common Error:

Poor SIMPLIFY reasoning: Students correctly distribute to get \(4\mathrm{y} - 7 - \mathrm{y} - 5\), but then make arithmetic errors when combining constants.

They might calculate \(-7 - 5\) incorrectly, perhaps getting \(-2\) instead of \(-12\), or even \(+12\) instead of \(-12\).

This leads to confusion and potentially selecting Choice (A) \(3\mathrm{y} - 16\) or Choice (D) \(3\mathrm{y} + 2\)

The Bottom Line:

This problem tests your precision with signs and your systematic approach to algebraic simplification. The key is treating subtraction of parentheses like distributing a negative one: \(-(\mathrm{y} + 5) = -\mathrm{y} - 5\).