Which expression is equivalent to 34x + 34y?

1. TRANSLATE the problem information

• Given expression: \(34\mathrm{x} + 34\mathrm{y}\)
• Need to find: An equivalent expression from the choices

2. INFER the solution strategy

• Both terms have the same coefficient: 34
• This suggests we can factor out the common factor
• Strategy: Use the distributive property in reverse

3. SIMPLIFY by factoring out the common factor

• Factor out 34 from both terms:

\(34\mathrm{x} + 34\mathrm{y} = 34(\mathrm{x} + \mathrm{y})\)

4. INFER which answer choice matches

• Choice A: \(34\mathrm{xy}\) means 34 times x times y (multiplication, not addition)
• Choice B: \(34(\mathrm{x} + \mathrm{y})\) exactly matches our result
• Choice C: \(68\mathrm{y} = 34\mathrm{y} + 34\mathrm{y}\) (wrong variables)
• Choice D: \(68\mathrm{x} = 34\mathrm{x} + 34\mathrm{x}\) (wrong variables)

Answer: B. \(34(\mathrm{x} + \mathrm{y})\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they can factor out common coefficients, or they confuse addition of terms with multiplication of variables.

Some students see \(34\mathrm{x} + 34\mathrm{y}\) and think it means "34 times x plus 34 times y equals 34 times xy" and incorrectly select Choice A (\(34\mathrm{xy}\)). This shows confusion between adding terms versus multiplying variables.

Second Most Common Error:

Conceptual confusion about combining like terms: Students incorrectly think that \(34\mathrm{x} + 34\mathrm{y}\) means adding the coefficients to get 68, then randomly assign this to either x or y.

This may lead them to select Choice C (\(68\mathrm{y}\)) or Choice D (\(68\mathrm{x}\)), not understanding that x and y are different variables that cannot be combined this way.

The Bottom Line:

This problem tests whether students understand the fundamental difference between factoring out common factors versus other algebraic operations. The key insight is recognizing when the distributive property can be applied in reverse.