How many solutions does the equation \(\frac{\mathrm{y}+6}{2} = \frac{3\mathrm{y}}{2} - (\mathrm{y}-1)\) have?

1. SIMPLIFY by eliminating fractions

• Multiply every term by 2 (the LCD of the fractions):
  • \(2 \cdot \frac{y+6}{2} = 2 \cdot \frac{3y}{2} - 2 \cdot (y-1)\)
  • This gives us: \(y + 6 = 3y - 2(y-1)\)

2. SIMPLIFY the right side using distribution

• Apply distributive property to \(-2(y-1)\):
  • \(-2(y-1) = -2y + 2\)
  • Now we have: \(y + 6 = 3y - 2y + 2\)

3. SIMPLIFY by combining like terms

• Combine \(3y - 2y\) on the right side:
  • \(y + 6 = y + 2\)

4. SIMPLIFY further by isolating constants

• Subtract \(y\) from both sides:
  • \(6 = 2\)

5. INFER the meaning of this result

• The statement \(6 = 2\) is never true - it's a contradiction
• When solving an equation leads to a contradiction, the original equation has no solutions

Answer: D) Zero

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when distributing the negative, particularly with \(-2(y-1)\). They might incorrectly get \(-2y - 2\) instead of \(-2y + 2\), leading to a different final equation like \(y + 6 = y - 2\), which would give \(6 = -2\) (still a contradiction, but they might not recognize it correctly).

This computational error can cause confusion about what the final result means, leading them to select Choice A (Exactly one) if they think they made a simple mistake.

Second Most Common Error:

Poor INFER reasoning: Students correctly reach \(6 = 2\) but don't understand what a contradiction means in the context of equation solving. They might think this means there's some solution they missed or that they should try a different approach.

This conceptual gap leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can execute a complete algebraic simplification AND interpret what contradictions mean. Many students can do the algebra but struggle with the conceptual leap that "impossible statement = no solutions."