y = 5x + 4 y = 5x^2 + 4 Which ordered pair \(\mathrm{(x, y)}\) is a solution to the...

1. TRANSLATE the problem information

• Given system of equations:
  • \(\mathrm{y = 5x + 4}\) (linear equation)
  • \(\mathrm{y = 5x^2 + 4}\) (quadratic equation)
• Need to find: Ordered pair (x, y) that satisfies both equations

2. INFER the solution strategy

• Since both equations equal y, the intersection occurs where the right-hand sides are equal
• Set the expressions equal: \(\mathrm{5x + 4 = 5x^2 + 4}\)
• This creates a single equation in x that we can solve

3. SIMPLIFY the equation

• Start with: \(\mathrm{5x + 4 = 5x^2 + 4}\)
• Subtract 4 from both sides: \(\mathrm{5x = 5x^2}\)
• Rearrange to standard form: \(\mathrm{5x^2 - 5x = 0}\)
• Factor out the common factor 5x: \(\mathrm{5x(x - 1) = 0}\)

4. APPLY zero product property

• If \(\mathrm{5x(x - 1) = 0}\), then either:
  • \(\mathrm{5x = 0}\), which gives \(\mathrm{x = 0}\)
  • \(\mathrm{x - 1 = 0}\), which gives \(\mathrm{x = 1}\)

5. SUBSTITUTE back to find y-coordinates

• When \(\mathrm{x = 0}\): \(\mathrm{y = 5(0) + 4 = 4}\) → Solution: \(\mathrm{(0, 4)}\)
• When \(\mathrm{x = 1}\): \(\mathrm{y = 5(1) + 4 = 9}\) → Solution: \(\mathrm{(1, 9)}\)

6. INFER which answer choice matches

• Looking at the choices, \(\mathrm{(0, 4)}\) corresponds to choice B

Answer: B. (0, 4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they should set the two expressions for y equal to each other. Instead, they might try to substitute answer choices one by one, or attempt to solve each equation separately without connecting them. This leads to confusion about how to approach the system and often results in guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic mistakes during the manipulation steps. Common errors include:

• Sign errors when moving terms (getting \(\mathrm{5x^2 + 5x = 0}\) instead of \(\mathrm{5x^2 - 5x = 0}\))
• Forgetting to factor out the common factor
• Arithmetic errors when substituting back

This may lead them to find incorrect x-values and subsequently select wrong answer choices.

The Bottom Line:

This problem requires recognizing the key insight that equal expressions can be set equal to each other. Students who miss this strategic approach often struggle to make progress systematically and resort to trial-and-error with the answer choices.