Question:If y = 3x - 4, which of the following expressions is equivalent to 2y + 8?6x6x - 86x +...

1. INFER the solution approach

• We have \(\mathrm{y = 3x - 4}\) and need to find what \(\mathrm{2y + 8}\) equals
• Key insight: Replace y with its equivalent expression \(\mathrm{(3x - 4)}\) in the target expression
• This will give us an expression in terms of x only

2. TRANSLATE the substitution

• Start with: \(\mathrm{2y + 8}\)
• Substitute \(\mathrm{y = 3x - 4}\): \(\mathrm{2(3x - 4) + 8}\)
• Now we have everything in terms of x

3. SIMPLIFY using distributive property

• Apply distributive property to \(\mathrm{2(3x - 4)}\):
  \(\mathrm{2(3x - 4) = 2(3x) + 2(-4) = 6x - 8}\)
• So our expression becomes: \(\mathrm{6x - 8 + 8}\)

4. SIMPLIFY by combining like terms

• Combine the constants: \(\mathrm{-8 + 8 = 0}\)
• Result: \(\mathrm{6x - 8 + 8 = 6x + 0 = 6x}\)

Answer: A (\(\mathrm{6x}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly substitute but make an error with the distributive property or forget to add the final \(\mathrm{+8}\).

For example, they might write \(\mathrm{2(3x - 4) + 8 = 6x - 8}\) and stop there, forgetting to add the \(\mathrm{+8}\). This leads them to select Choice B (\(\mathrm{6x - 8}\)).

Alternatively, they might incorrectly distribute: \(\mathrm{2(3x - 4) = 6x - 4}\), then add 8 to get \(\mathrm{6x + 4}\), which isn't among the choices, causing confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize they need to substitute the expression for y, instead trying to manipulate \(\mathrm{2y + 8}\) algebraically without using the given information.

This causes them to get stuck immediately since they can't simplify \(\mathrm{2y + 8}\) without knowing what y equals, leading to abandoning systematic solution and guessing.

The Bottom Line:

This problem tests whether students can execute a fundamental algebraic strategy (substitution) followed by careful algebraic manipulation. The key is recognizing that substitution is necessary and then being methodical with the distributive property and combining like terms.