Question:p = 5p^2 - 3q = 1The solution to the given system of equations is \(\mathrm{(p,q)}\). What is the value...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{p = 5}\) (first equation)
  • \(\mathrm{p^2 - 3q = 1}\) (second equation)
• What we need to find: the value of q

2. INFER the solution strategy

• Since we already know \(\mathrm{p = 5}\), we can substitute this value directly into the second equation
• This will give us a linear equation in q that we can solve

3. SIMPLIFY through substitution

• Replace p with 5 in the equation \(\mathrm{p^2 - 3q = 1}\):
  \(\mathrm{(5)^2 - 3q = 1}\)
• Calculate the exponent:
  \(\mathrm{25 - 3q = 1}\)

4. SIMPLIFY to solve for q

• Move 25 to the right side:
  \(\mathrm{-3q = 1 - 25}\)
• Simplify:
  \(\mathrm{-3q = -24}\)
• Divide both sides by -3:
  \(\mathrm{q = 8}\)

Answer: 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make arithmetic errors when working with negative numbers, especially when dividing -24 by -3. They might incorrectly get \(\mathrm{q = -8}\) instead of \(\mathrm{q = 8}\), forgetting that dividing two negative numbers gives a positive result.

This leads to confusion since 8 isn't among their calculated options, causing them to second-guess their work or randomly select an answer.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might misunderstand the problem setup and think they need to solve both equations simultaneously, rather than recognizing that \(\mathrm{p = 5}\) is already solved and can be directly substituted. This leads them to overcomplicate a straightforward substitution problem.

This causes them to get stuck trying to use elimination or other complex methods, leading to confusion and guessing.

The Bottom Line:

This is fundamentally a substitution problem disguised as a system. The key insight is recognizing that when one variable is already isolated (\(\mathrm{p = 5}\)), you simply substitute and solve for the remaining variable.