a = 6b = sqrt(43 + a)A solution to the given system of equations is \(\mathrm{(a, b)}\). What is the...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{a = 6}\)
  • \(\mathrm{b = \sqrt{43 + a}}\)
  • Need to find: \(\mathrm{ab}\)

2. INFER the solution approach

• Since we have a direct value for a and b is expressed in terms of a, substitute the known value of a into the equation for b
• After finding b, multiply a and b to get the final answer

3. SIMPLIFY by substituting and calculating

• Substitute \(\mathrm{a = 6}\) into the equation for b:
  \(\mathrm{b = \sqrt{43 + 6}}\)
• Perform the addition inside the square root:
  \(\mathrm{b = \sqrt{49}}\)
• Calculate the square root:
  \(\mathrm{b = 7}\)
• Find the product:
  \(\mathrm{ab = 6 \times 7 = 42}\)

Answer: C. 42

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making calculation errors with the square root

Students might incorrectly calculate \(\mathrm{\sqrt{49}}\), perhaps thinking it equals 8 or 6 instead of 7. This could happen if they confuse perfect squares or make mental arithmetic errors.

This may lead them to select Choice D (48) if they think \(\mathrm{b = 8}\), or other incorrect choices.

Second Most Common Error:

Incomplete solution process: Finding b correctly but then adding instead of multiplying

Some students correctly find \(\mathrm{b = 7}\) but then calculate \(\mathrm{a + b = 6 + 7 = 13}\) instead of \(\mathrm{ab = 6 \times 7 = 42}\).

This may lead them to select Choice A (13).

The Bottom Line:

This problem tests careful arithmetic execution more than complex mathematical reasoning. Success depends on accurate calculation of the square root and remembering that the question asks for the product \(\mathrm{ab}\), not the sum.