5x + y = 14 x^2 = y + 10 The graphs of the equations in the given system of...

1. TRANSLATE the problem information

• Given system:
  • \(5\mathrm{x} + \mathrm{y} = 14\) (linear equation)
  • \(\mathrm{x}^2 = \mathrm{y} + 10\) (quadratic equation)
• Find: Possible value(s) of x where the graphs intersect

2. INFER the solution approach

• Since we have one linear equation in x and y, substitution is the most efficient method
• Solve the linear equation for y, then substitute into the quadratic equation
• This will give us a quadratic equation in x only

3. SIMPLIFY to isolate y in the first equation

• From \(5\mathrm{x} + \mathrm{y} = 14\)
• Subtract 5x from both sides: \(\mathrm{y} = 14 - 5\mathrm{x}\)

4. SIMPLIFY by substituting into the second equation

• Replace y in \(\mathrm{x}^2 = \mathrm{y} + 10\):
• \(\mathrm{x}^2 = (14 - 5\mathrm{x}) + 10\)
• \(\mathrm{x}^2 = 24 - 5\mathrm{x}\)
• Move all terms to one side: \(\mathrm{x}^2 + 5\mathrm{x} - 24 = 0\)

5. SIMPLIFY by factoring the quadratic

• Need two numbers that multiply to -24 and add to 5
• Try factor pairs of 24: 8 and -3 work since \(8(-3) = -24\) and \(8 + (-3) = 5\)
• Factor: \((\mathrm{x} + 8)(\mathrm{x} - 3) = 0\)

6. APPLY CONSTRAINTS to find solutions

• By zero product property: \(\mathrm{x} + 8 = 0\) or \(\mathrm{x} - 3 = 0\)
• Therefore: \(\mathrm{x} = -8\) or \(\mathrm{x} = 3\)
• Both are mathematically valid intersection points

7. APPLY CONSTRAINTS to select from answer choices

• Looking at the given options: (A) -8, (B) -5, (C) -3, (D) 5
• \(\mathrm{x} = -8\) appears as choice (A)
• \(\mathrm{x} = 3\) does not appear in the choices

Answer: (A) -8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may attempt to solve both equations simultaneously using elimination instead of recognizing that substitution is more direct when one equation is already linear in one variable.

This leads to unnecessary complexity and potential arithmetic errors, causing confusion and possibly guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when expanding \((14 - 5\mathrm{x}) + 10\) or when rearranging to standard form, leading to an incorrect quadratic equation.

For example, getting \(\mathrm{x}^2 + 5\mathrm{x} + 24 = 0\) instead of \(\mathrm{x}^2 + 5\mathrm{x} - 24 = 0\), which would have no real solutions and cause them to get stuck and guess.

The Bottom Line:

This problem tests whether students can strategically choose substitution over elimination and execute multi-step algebraic simplification accurately. The key insight is recognizing that having one linear equation makes substitution the natural choice.