The system of equations below represents the intersection of two curves in the xy-plane.3x + y = 113x^2 = y...

1. TRANSLATE the problem information

• Given system:
  • \(3x + y = 11\) (linear equation)
  • \(3x² = y + 79\) (quadratic equation)
• We need to find possible x-values where these curves intersect

2. INFER the solution strategy

• Since we have two equations with two unknowns, we can solve using substitution
• The first equation is already solved for y in terms of x, making substitution straightforward
• We'll substitute this expression into the second equation to get one equation with one variable

3. SIMPLIFY by substitution

• From equation 1: \(y = 11 - 3x\)
• Substitute into equation 2:
  \(3x² = (11 - 3x) + 79\)
• Combine like terms:
  \(3x² = 90 - 3x\)
• Move all terms to one side:
  \(3x² + 3x - 90 = 0\)

4. SIMPLIFY the quadratic equation

• Divide everything by 3:
  \(x² + x - 30 = 0\)
• Factor by finding two numbers that multiply to -30 and add to 1:
  Those numbers are 6 and -5
• Factor: \((x + 6)(x - 5) = 0\)

5. INFER the solutions using zero product property

• If \((x + 6)(x - 5) = 0\), then either:
  • \(x + 6 = 0\), so \(x = -6\), OR
  • \(x - 5 = 0\), so \(x = 5\)

6. APPLY CONSTRAINTS to select from answer choices

• Both \(x = -6\) and \(x = 5\) are mathematically valid intersection points
• Looking at the given choices: (A) -6, (B) -5, (C) 3, (D) 6
• Only \(x = -6\) appears among the options

Answer: (A) -6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may try to solve the system using elimination instead of substitution, leading to unnecessary complications with the quadratic term. They might attempt to multiply the first equation by 3 to match coefficients, but this doesn't help eliminate the x² term in the second equation. This leads to confusion and abandoning the systematic approach, causing them to guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students may make algebraic errors when expanding and combining terms after substitution. For example, they might incorrectly combine (11 - 3x) + 79 as 90 + 3x instead of 90 - 3x, leading to the wrong quadratic equation \(x² - x - 30 = 0\). Factoring this incorrectly gives solutions that don't match any answer choice, leading to confusion and guessing.

The Bottom Line:

This problem requires students to confidently navigate the transition from a system of equations to a single quadratic equation, then successfully factor and apply answer choice constraints. The key insight is recognizing substitution as the most efficient path forward.