x + 7 = 10\(\mathrm{(x + 7)^2 = y}\)Which ordered pair \(\mathrm{(x, y)}\) is a solution to the given system...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{x + 7 = 10}\)
  • \(\mathrm{(x + 7)^2 = y}\)
• Find: The ordered pair (x, y) that satisfies both equations

2. INFER the most efficient approach

• Key insight: The first equation tells us that \(\mathrm{x + 7 = 10}\)
• Rather than solving for x immediately, we can substitute this entire relationship into the second equation
• This means \(\mathrm{(x + 7)^2}\) becomes \(\mathrm{(10)^2}\) in the second equation

3. SIMPLIFY to find y

• Substitute \(\mathrm{x + 7 = 10}\) into the second equation:
  \(\mathrm{(10)^2 = y}\)
• Calculate: \(\mathrm{y = 100}\)

4. SIMPLIFY to find x

• From the first equation \(\mathrm{x + 7 = 10}\):
  \(\mathrm{x = 10 - 7 = 3}\)

Answer: A. \(\mathrm{(3, 100)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the substitution opportunity and instead try to work with individual variables rather than expressions.

After finding \(\mathrm{x = 3}\), they might substitute incorrectly into the second equation, thinking they need \(\mathrm{(x)^2 = y}\), giving them \(\mathrm{y = 3^2 = 9}\). Since this isn't an option, they get confused. Alternatively, they might think \(\mathrm{y = x + 7 = 10}\), leading them to select Choice C. \(\mathrm{(3, 10)}\).

Second Most Common Error:

Conceptual confusion about systems: Students might think that once they find \(\mathrm{x = 3}\), the y-value should somehow relate directly to x rather than following from the second equation.

This oversimplified thinking might lead them to assume \(\mathrm{y = x}\), causing them to select Choice B. \(\mathrm{(3, 3)}\).

The Bottom Line:

The key to this problem is recognizing that expressions (like \(\mathrm{x + 7}\)) can be substituted as units, not just individual variables. Students who miss this insight make the problem much harder than it needs to be.