If y = x + 3, which of the following is equivalent to 2y - 6?

1. TRANSLATE the problem setup

• Given information: \(\mathrm{y = x + 3}\)
• Need to find: what \(\mathrm{2y - 6}\) equals when we substitute the given expression for y

2. TRANSLATE into mathematical operations

• Substitute the expression \(\mathrm{(x + 3)}\) everywhere you see y in \(\mathrm{2y - 6}\)
• This gives us: \(\mathrm{2(x + 3) - 6}\)

3. SIMPLIFY using the distributive property

• Apply \(\mathrm{a(b + c) = ab + ac}\) to get: \(\mathrm{2(x + 3) = 2x + 6}\)
• So we have: \(\mathrm{2x + 6 - 6}\)

4. SIMPLIFY by combining like terms

• The +6 and -6 cancel out: \(\mathrm{2x + 6 - 6 = 2x}\)

Answer: B (\(\mathrm{2x}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students substitute incorrectly, replacing y with just x instead of the complete expression \(\mathrm{(x + 3)}\).

If they think "\(\mathrm{y = x + 3}\)" means "\(\mathrm{y}\) equals \(\mathrm{x}\)," they substitute \(\mathrm{y = x}\) into \(\mathrm{2y - 6}\) to get \(\mathrm{2x - 6}\).
This leads them to select Choice A (\(\mathrm{2x - 6}\)).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students apply the distributive property correctly to get \(\mathrm{2x + 6 - 6}\), but then forget to complete the simplification by combining like terms.

They might stop at \(\mathrm{2x + 6}\) after distributing, forgetting about the -6 part of the original expression.
This may lead them to select Choice C (\(\mathrm{2x + 6}\)).

The Bottom Line:

This problem tests whether students can carefully execute a complete substitution (replacing the entire variable with the entire given expression) and then follow through with all algebraic simplification steps without dropping terms along the way.