What is the solution (x, y) to the given system of equations? y = 2x + 3 x = 1...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{y = 2x + 3}\)
  • \(\mathrm{x = 1}\)
• Find: The solution as an ordered pair \(\mathrm{(x, y)}\)

2. INFER the solution strategy

• Since \(\mathrm{x = 1}\) is already given, we don't need elimination or complex substitution
• We can directly substitute \(\mathrm{x = 1}\) into the first equation to find \(\mathrm{y}\)

3. SIMPLIFY by substituting and calculating

• Substitute \(\mathrm{x = 1}\) into \(\mathrm{y = 2x + 3}\):
  \(\mathrm{y = 2(1) + 3}\)
  \(\mathrm{y = 2 + 3}\)
  \(\mathrm{y = 5}\)

4. TRANSLATE the result to ordered pair form

• The solution is \(\mathrm{(x, y) = (1, 5)}\)

Answer: B. (1, 5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making arithmetic errors during substitution

Students correctly identify that they need to substitute \(\mathrm{x = 1}\), but make calculation mistakes:

• \(\mathrm{y = 2(1) + 3 = 2}\) (forgetting to add 3)
• This leads them to select Choice A. (1, 2)

Second Most Common Error:

Poor TRANSLATE reasoning: Misreading or misunderstanding which value represents x

Some students might confuse themselves about which variable has the given value, potentially using \(\mathrm{x = 2}\) instead of \(\mathrm{x = 1}\):

• If \(\mathrm{x = 2}\): \(\mathrm{y = 2(2) + 3 = 7}\)
• This leads them to select Choice D. (2, 7)

The Bottom Line:

This problem tests careful reading and basic substitution skills. The key insight is recognizing that when one variable is given directly, the solution becomes a simple one-step substitution rather than a complex system-solving process.