The solution to the given system of equations is (x, y).\(\mathrm{y = (x - 1)(x - 8)}\)x + 7 =...

1. INFER the solving strategy

• Given information:
  • \(\mathrm{y = (x - 1)(x - 8)}\) [quadratic equation]
  • \(\mathrm{x + 7 = 3}\) [linear equation]

• The key insight: Since we have one equation that gives us x directly, solve that first, then substitute.

2. SIMPLIFY the linear equation to find x

• Start with the simpler equation: \(\mathrm{x + 7 = 3}\)
• Subtract 7 from both sides: \(\mathrm{x = 3 - 7 = -4}\)

3. SIMPLIFY by substituting x = -4 into the quadratic equation

• Replace x with -4 in \(\mathrm{y = (x - 1)(x - 8)}\)
• \(\mathrm{y = (-4 - 1)(-4 - 8)}\)
• \(\mathrm{y = (-5)(-12)}\)
• When multiplying two negative numbers, the result is positive: \(\mathrm{y = 60}\)

Answer: C) 60

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when working with negative numbers, particularly when substituting \(\mathrm{x = -4}\).

For example, they might calculate \(\mathrm{(-4 - 1)}\) as -3 instead of -5, or they might incorrectly handle the multiplication of negative numbers, getting \(\mathrm{-60}\) instead of \(\mathrm{+60}\). This leads them to select Choice A (-60).

Second Most Common Error:

Poor INFER reasoning: Students try to solve the quadratic equation first instead of recognizing that the linear equation gives them x directly.

This leads to unnecessary complexity, potential algebra errors, and confusion about which equation to use first. Students may become overwhelmed and resort to guessing among the answer choices.

The Bottom Line:

This problem tests whether students can identify the most efficient solution path (solve the simple equation first) and execute arithmetic with negative numbers accurately. The key is recognizing that systems don't always require complex elimination or substitution - sometimes one equation directly gives you what you need.