Given the equation 6 - 4y = -2, which equation has the same solution as the given equation?4y = -84y...

1. TRANSLATE the problem information

• Given equation: \(6 - 4\mathrm{y} = -2\)
• Need to find: Which answer choice has the same solution
• What this means: Find the equation that gives the same y-value

2. INFER the most efficient approach

• 'Same solution' means both equations have the same y-value when solved
• Two strategies: solve for y first, then check choices OR transform the original equation to match a choice
• Let's use algebraic transformation since it's more direct

3. SIMPLIFY through algebraic steps

• Start with: \(6 - 4\mathrm{y} = -2\)
• Subtract 6 from both sides: \(6 - 6 - 4\mathrm{y} = -2 - 6\)
• This gives us: \(-4\mathrm{y} = -8\)
• Multiply both sides by -1: \(4\mathrm{y} = 8\)

4. INFER the final answer

• Our transformed equation is \(4\mathrm{y} = 8\)
• This directly matches choice (D)
• We can verify: if \(4\mathrm{y} = 8\), then \(\mathrm{y} = 2\)
• Check original: \(6 - 4(2) = 6 - 8 = -2\) ✓

Answer: D (\(4\mathrm{y} = 8\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign confusion when subtracting 6 from both sides

Students might write: \(6 - 4\mathrm{y} = -2\), then \(-4\mathrm{y} = -2 + 6 = 4\), leading to \(4\mathrm{y} = -4\).

This stems from incorrectly thinking 'subtract 6' means 'add 6 to the right side.' This may lead them to select Choice (A) (\(4\mathrm{y} = -8\)) after additional sign errors, or causes confusion and guessing.

Second Most Common Error:

Missing conceptual understanding: Not grasping what 'same solution' means

Students might think they need to find an equation that looks similar to the original, rather than one that yields the same y-value. This leads to random selection without systematic solving.

The Bottom Line:

This problem tests careful algebraic manipulation with negative numbers. The key insight is that maintaining equality while transforming equations requires applying the same operation to both sides, with particular attention to sign changes.