Which expression is equivalent to \((-4y^2 + 15) + (7y^2 - 8)\)?-{11y^2 + 7}3y^2 + 73y^2 + 2311y^2 + 7

1. TRANSLATE the problem information

• Given: \((-4y^2 + 15) + (7y^2 - 8)\)
• Need to find: An equivalent expression (simplified form)
• This means combining like terms to get the simplest form

2. SIMPLIFY by removing parentheses

• \((-4y^2 + 15) + (7y^2 - 8)\)
• \(= -4y^2 + 15 + 7y^2 - 8\)
• The first set of parentheses doesn't change signs
• The second set: \(+(7y^2 - 8) = +7y^2 - 8\)

3. SIMPLIFY by grouping like terms

• \(= (-4y^2 + 7y^2) + (15 - 8)\)
• Group the \(y^2\) terms together: \(-4y^2 + 7y^2\)
• Group the constant terms together: \(15 - 8\)

4. SIMPLIFY by combining like terms

• \(y^2\) terms: \(-4y^2 + 7y^2 = 3y^2\)
• Constant terms: \(15 - 8 = 7\)
• Final result: \(3y^2 + 7\)

Answer: B) \(3y^2 + 7\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students correctly identify that they need to combine like terms, but make a sign error with the constant terms.

They see 15 and 8, and since addition feels more natural than subtraction, they compute \(15 + 8 = 23\) instead of \(15 - 8 = 7\). The \(y^2\) terms are handled correctly: \(-4y^2 + 7y^2 = 3y^2\).

This leads them to select Choice C (\(3y^2 + 23\)).

Second Most Common Error:

Weak SIMPLIFY skill: Students make a sign error when combining the \(y^2\) terms, treating \(-4y^2\) as if it were \(+4y^2\).

They calculate the \(y^2\) coefficient as \(4 + 7 = 11\) instead of \(-4 + 7 = 3\), while correctly handling the constants: \(15 - 8 = 7\).

This leads them to select Choice D (\(11y^2 + 7\)).

The Bottom Line:

The key challenge is maintaining accuracy with positive and negative signs throughout multiple algebraic steps, especially when dealing with subtraction that appears after removing parentheses.