Question:Consider the expression \(6 - 2(\mathrm{x} - 5)\).It is rewritten in the form ax + b, where a and b...

1. TRANSLATE the problem requirements

• Given: Expression \(6 - 2(\mathrm{x} - 5)\)
• Task: Rewrite in the form \(\mathrm{ax} + \mathrm{b}\) and find the value of \(\mathrm{a}\)
• What this tells us: We need to expand and simplify to get the coefficient of \(\mathrm{x}\)

2. SIMPLIFY using the distributive property

• Apply \(-2\) to both terms inside the parentheses:

\(-2(\mathrm{x} - 5) = -2 \times \mathrm{x} + (-2) \times (-5)\)

\(= -2\mathrm{x} + 10\)

• Be careful with signs: \(-2\) times \(-5\) gives positive \(10\)

3. SIMPLIFY by substituting and combining like terms

• Substitute back:

\(6 - 2(\mathrm{x} - 5) = 6 - 2\mathrm{x} + 10\)

• Combine constant terms:

\(6 + 10 = 16\)

• Final form: \(-2\mathrm{x} + 16\)

4. TRANSLATE to identify the coefficient

• In the form \(\mathrm{ax} + \mathrm{b}\), we have \(-2\mathrm{x} + 16\)
• The coefficient of \(\mathrm{x}\) is \(\mathrm{a} = -2\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students may not recognize that they need to distribute before identifying the coefficient, thinking that since no \(\mathrm{x}\) term is explicitly visible in \(6 - 2(\mathrm{x} - 5)\), the coefficient of \(\mathrm{x}\) must be zero.

This reasoning leads them to ignore the distributive step entirely and conclude that \(\mathrm{a} = 0\).

This may lead them to select Choice C (0).

Second Most Common Error:

SIMPLIFY execution error: Students correctly understand they need to distribute but make a sign error when reading or copying the original expression, treating it as \(6 + 2(\mathrm{x} - 5)\) instead of \(6 - 2(\mathrm{x} - 5)\).

Following correct distribution steps:

\(6 + 2(\mathrm{x} - 5) = 6 + 2\mathrm{x} - 10\)

\(= 2\mathrm{x} - 4\)

giving them \(\mathrm{a} = 2\).

This may lead them to select Choice D (2).

The Bottom Line:

This problem tests whether students can systematically apply the distributive property with negative numbers and recognize that algebraic expressions need to be fully simplified before identifying coefficients. The key insight is that the coefficient isn't visible until after distribution and combining like terms.