y = x + 9 y = x^2 + 16x + 63 A solution to the given system of equations...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{y = x + 9}\) (linear equation)
  • \(\mathrm{y = x^2 + 16x + 63}\) (quadratic equation)
• Find: Greatest possible value of x

2. INFER the solution approach

• Since both expressions equal y, I can use substitution
• Set the two expressions equal: \(\mathrm{x + 9 = x^2 + 16x + 63}\)
• This eliminates y and gives us one equation in one variable

3. SIMPLIFY to standard quadratic form

• Start with: \(\mathrm{x + 9 = x^2 + 16x + 63}\)
• Move all terms to one side: \(\mathrm{0 = x^2 + 16x + 63 - x - 9}\)
• Combine like terms: \(\mathrm{0 = x^2 + 15x + 54}\)

4. SIMPLIFY by factoring the quadratic

• Need two numbers that multiply to 54 and add to 15
• Try factor pairs of 54: \(\mathrm{6 \times 9 = 54}\) and \(\mathrm{6 + 9 = 15}\) ✓
• Factor: \(\mathrm{0 = (x + 6)(x + 9)}\)

5. APPLY CONSTRAINTS using zero product property

• If \(\mathrm{(x + 6)(x + 9) = 0}\), then \(\mathrm{x + 6 = 0}\) or \(\mathrm{x + 9 = 0}\)
• Solving: \(\mathrm{x = -6}\) or \(\mathrm{x = -9}\)
• Since we want the greatest possible value: \(\mathrm{x = -6}\)

Answer: A. -6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students make arithmetic errors when rearranging the equation or combining like terms.

For example, they might incorrectly write \(\mathrm{x + 9 = x^2 + 16x + 63}\) as \(\mathrm{0 = x^2 + 17x + 54}\) instead of \(\mathrm{0 = x^2 + 15x + 54}\). This leads to attempting to factor the wrong quadratic expression, making it impossible to find integer solutions. This leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students find both solutions (-6 and -9) correctly but select the smaller value instead of the greater one.

They solve the system correctly and get \(\mathrm{x = -6}\) or \(\mathrm{x = -9}\), but misread "greatest possible value" and choose -9 because they think of it as the "larger number" without considering that \(\mathrm{-6 \gt -9}\). This may lead them to select Choice D (63) if they get confused, or they might not see -9 as an option and guess.

The Bottom Line:

This problem combines algebraic manipulation skills with careful reading of what's being asked. Students need to stay organized during the algebraic work and remember that "greatest" means the largest value on the number line.