The solution to the given system of equations is (x, y). What is the value of y?y = x^2 +...

1. INFER the most efficient solution approach

• Given information:
  • \(\mathrm{y = x² + 14x + 48}\) (quadratic equation)
  • \(\mathrm{x + 8 = 11}\) (linear equation)
  • Need to find the value of y

• Strategy insight: Since one equation is much simpler (linear), solve that first to find x, then substitute into the other equation.

2. SIMPLIFY to find the value of x

• Start with the linear equation: \(\mathrm{x + 8 = 11}\)
• Subtract 8 from both sides: \(\mathrm{x = 11 - 8}\)
• Therefore: \(\mathrm{x = 3}\)

3. SIMPLIFY to find the value of y using substitution

• Substitute \(\mathrm{x = 3}\) into the quadratic equation: \(\mathrm{y = x² + 14x + 48}\)
• Replace x with 3: \(\mathrm{y = (3)² + 14(3) + 48}\)
• Calculate step by step:
  • \(\mathrm{(3)² = 9}\)
  • \(\mathrm{14(3) = 42}\)
  • \(\mathrm{y = 9 + 42 + 48 = 99}\)

Answer: 99

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic mistakes when evaluating \(\mathrm{y = (3)² + 14(3) + 48}\)

Common calculation errors include:

• Forgetting that \(\mathrm{(3)² = 9}\), not 6
• Incorrectly calculating \(\mathrm{14(3)}\) as something other than 42
• Making addition errors in the final step: \(\mathrm{9 + 42 + 48}\)

This leads to selecting an incorrect numerical answer or abandoning the problem due to confusion.

Second Most Common Error:

Poor INFER reasoning: Students try to solve both equations simultaneously using elimination method instead of recognizing the simpler substitution approach

They might attempt to manipulate \(\mathrm{y = x² + 14x + 48}\) and \(\mathrm{x + 8 = 11}\) to eliminate variables, creating unnecessary complexity. This leads to confusion and potentially giving up on the systematic solution.

The Bottom Line:

This problem tests whether students can recognize when substitution is the most efficient method and execute basic algebraic operations accurately. The key insight is choosing the right strategy first, then executing it carefully.