x = 3\(\mathrm{y = (15 - x)^2}\)A solution to the given system of equations is \(\mathrm{(x, y)}\). What is the...

1. TRANSLATE the problem information

• Given: A system with \(\mathrm{x = 3}\) and \(\mathrm{y = (15 - x)^2}\)
• Find: The value of \(\mathrm{xy}\) where \(\mathrm{(x, y)}\) is the solution

2. INFER the solution strategy

• Since the first equation directly gives us \(\mathrm{x = 3}\), we can substitute this value into the second equation to find \(\mathrm{y}\)
• Once we have both \(\mathrm{x}\) and \(\mathrm{y}\), we multiply them together

3. SIMPLIFY to find y

• Substitute \(\mathrm{x = 3}\) into \(\mathrm{y = (15 - x)^2}\):
• \(\mathrm{y = (15 - 3)^2}\)
• \(\mathrm{y = 12^2}\)
• \(\mathrm{y = 144}\)

4. SIMPLIFY to find xy

• \(\mathrm{xy = 3 \times 144 = 432}\)

Answer: A. 432

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misreading the second equation as \(\mathrm{y = 15 - x}\) instead of \(\mathrm{y = (15 - x)^2}\)

Students miss the square and calculate: \(\mathrm{y = 15 - 3 = 12}\), leading to \(\mathrm{xy = 3 \times 12 = 36}\). When 36 isn't among the choices, they might guess or recalculate incorrectly, potentially selecting Choice C (45) if they mistakenly think \(\mathrm{y = 15}\).

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors in the calculation phase

Students correctly set up \(\mathrm{y = (15 - 3)^2}\) but then calculate incorrectly. Common mistakes include forgetting that \(\mathrm{12^2 = 144}\) or making errors in the final multiplication \(\mathrm{3 \times 144}\). This leads to confusion and guessing among the available choices.

The Bottom Line:

This problem tests whether students can systematically work through substitution while maintaining accuracy in both reading mathematical notation (especially the square) and performing multi-step calculations.