Question:y = x^2 + 3x + 4y = 2x^2 + 3x - 5Which ordered pair \(\mathrm{(x, y)}\) is a solution...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{y = x^2 + 3x + 4}\) (first equation)
  • \(\mathrm{y = 2x^2 + 3x - 5}\) (second equation)
  • Need to find which ordered pair (x, y) satisfies both equations

2. INFER the solution strategy

• Since both equations are solved for y, we can set the right sides equal to each other
• This eliminates y and gives us one equation with only x
• Strategy: Set \(\mathrm{x^2 + 3x + 4 = 2x^2 + 3x - 5}\)

3. SIMPLIFY the equation to solve for x

• Start with: \(\mathrm{x^2 + 3x + 4 = 2x^2 + 3x - 5}\)
• Subtract 3x from both sides: \(\mathrm{x^2 + 4 = 2x^2 - 5}\)
• Subtract \(\mathrm{x^2}\) from both sides: \(\mathrm{4 = x^2 - 5}\)
• Add 5 to both sides: \(\mathrm{9 = x^2}\)
• Take the square root: \(\mathrm{x = ±3}\)

4. CONSIDER ALL CASES for the x-values

• We have \(\mathrm{x = 3}\) or \(\mathrm{x = -3}\)
• Need to find the corresponding y-values for each x

5. SIMPLIFY to find the y-values

• For \(\mathrm{x = 3}\):
  \(\mathrm{y = (3)^2 + 3(3) + 4}\)
  \(\mathrm{= 9 + 9 + 4}\)
  \(\mathrm{= 22}\)
• For \(\mathrm{x = -3}\):
  \(\mathrm{y = (-3)^2 + 3(-3) + 4}\)
  \(\mathrm{= 9 - 9 + 4}\)
  \(\mathrm{= 4}\)
• The two solutions are \(\mathrm{(3, 22)}\) and \(\mathrm{(-3, 4)}\)

6. APPLY CONSTRAINTS based on answer choices

• Looking at the given choices, only \(\mathrm{(3, 22)}\) appears as option D
• We can verify: both equations give \(\mathrm{y = 22}\) when \(\mathrm{x = 3}\)

Answer: D. (3, 22)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when combining like terms or moving terms between sides of the equation.

For example, when going from \(\mathrm{x^2 + 3x + 4 = 2x^2 + 3x - 5}\) to the next step, they might incorrectly handle the subtraction of 3x or make sign errors. A common mistake is getting \(\mathrm{x^2 + 9 = 2x^2}\) instead of \(\mathrm{9 = x^2}\), or forgetting to move all terms correctly. This leads to wrong x-values and may cause them to select Choice A (-3, 22) or get confused and guess.

Second Most Common Error:

Poor CONSIDER ALL CASES reasoning: Students find \(\mathrm{x = 3}\) but forget that \(\mathrm{x^2 = 9}\) also gives \(\mathrm{x = -3}\), so they only check one solution.

They might substitute \(\mathrm{x = 3}\) back into the equations correctly to get \(\mathrm{(3, 22)}\), but they don't realize there's another mathematically valid solution. While this doesn't affect getting the right answer since only \(\mathrm{(3, 22)}\) appears in the choices, it shows incomplete problem-solving and might lead to second-guessing their work.

The Bottom Line:

This problem tests whether students can systematically solve a system by elimination and handle the algebra carefully. The key insight is recognizing that setting equal expressions for y creates a solvable single-variable equation.