a - 2 = 14sqrt(a) = b, where a gt 0Which ordered pair \(\mathrm{(a, b)}\) is a solution to the...

1. TRANSLATE the problem information

• Given system of equations:
  • \(\mathrm{a - 2 = 14}\) (linear equation in a)
  • \(\sqrt{\mathrm{a}} = \mathrm{b}\) (square root equation relating a and b)
  • Constraint: \(\mathrm{a \gt 0}\)
• What we need to find: The ordered pair \(\mathrm{(a, b)}\) that satisfies both equations

2. SIMPLIFY the first equation to find a

• Start with: \(\mathrm{a - 2 = 14}\)
• Add 2 to both sides: \(\mathrm{a = 16}\)

3. SIMPLIFY the second equation to find b

• We know \(\mathrm{a = 16}\), so substitute into \(\sqrt{\mathrm{a}} = \mathrm{b}\)
• This gives us: \(\sqrt{16} = \mathrm{b}\)
• Evaluate the square root: \(\mathrm{b = 4}\)

4. Form the solution

• The ordered pair is \(\mathrm{(a, b) = (16, 4)}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when evaluating \(\sqrt{16}\)

Some students confuse square root with other operations or make basic calculation mistakes. They might think \(\sqrt{16} = 2\) (confusing with \(\sqrt{4}\)), or they might square 16 instead of taking its square root to get 256.

This may lead them to select Choice A \(\mathrm{(16, 2)}\) or Choice C \(\mathrm{(16, 256)}\)

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what b represents in the context

Some students might think that since the first equation gives 14, b should somehow equal 14, not recognizing that b is defined by the second equation \(\sqrt{\mathrm{a}} = \mathrm{b}\).

This may lead them to select Choice B \(\mathrm{(16, 14)}\)

The Bottom Line:

This problem tests your ability to work systematically through a simple system of equations. The key is recognizing that you solve for one variable first, then use that result in the second equation. Most errors come from rushing through the square root calculation or misunderstanding the relationship between the equations.