Point A is located at coordinates \((2, 5)\) and point B is located at coordinates \((14, \mathrm{y})\), where y gt...

1. TRANSLATE the problem information

• Given information:
  • Point A coordinates: \((2, 5)\)
  • Point B coordinates: \((14, y)\)
  • Distance from A to B: 13
  • Constraint: \(\mathrm{y \gt 5}\)
• What this tells us: We need to use the distance formula to create an equation we can solve for y

2. TRANSLATE the distance relationship into an equation

• The distance formula is: \(\mathrm{d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}}\)
• Substituting our values: \(\mathrm{13 = \sqrt{(14-2)^2 + (y-5)^2}}\)
• This gives us: \(\mathrm{13 = \sqrt{144 + (y-5)^2}}\)

3. SIMPLIFY to solve for y

• Square both sides to eliminate the square root: \(\mathrm{169 = 144 + (y-5)^2}\)
• Subtract 144 from both sides: \(\mathrm{25 = (y-5)^2}\)
• Take the square root: \(\mathrm{y-5 = ±5}\)
• This gives us two potential solutions: \(\mathrm{y = 10}\) or \(\mathrm{y = 0}\)

4. APPLY CONSTRAINTS to select the final answer

• Since the problem states \(\mathrm{y \gt 5}\), we must reject \(\mathrm{y = 0}\)
• Therefore: \(\mathrm{y = 10}\)

Answer: 10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS reasoning: Students solve the equation correctly but forget to check the constraint \(\mathrm{y \gt 5}\), leading them to consider both \(\mathrm{y = 10}\) and \(\mathrm{y = 0}\) as valid answers. They might arbitrarily pick \(\mathrm{y = 0}\) or get confused about which solution to choose, potentially entering 0 as their final answer.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when squaring both sides or solving \(\mathrm{(y-5)^2 = 25}\). Common mistakes include forgetting that squaring both sides gives 169 (not 13²), or not recognizing that taking the square root of 25 gives ±5. These calculation errors lead to incorrect values that don't satisfy the original equation.

The Bottom Line:

This problem tests whether students can systematically apply the distance formula and carefully work through constraint-based elimination of invalid solutions.