y = 3x 2x + y = 12 The solution to the given system of equations is \(\mathrm{(x, y)}\). What...

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\mathrm{y = 3x}\)
  • Second equation: \(\mathrm{2x + y = 12}\)
• What we need to find: The value of \(\mathrm{5x}\)

2. INFER the best approach

• Since the first equation already expresses y in terms of x, substitution is the most direct method
• We can substitute the expression for y directly into the second equation

3. SIMPLIFY by substituting and combining terms

• Substitute \(\mathrm{y = 3x}\) into \(\mathrm{2x + y = 12}\):
  \(\mathrm{2x + 3x = 12}\)
• Combine like terms: \(\mathrm{5x = 12}\)

4. INFER when you're finished

• The problem asks for \(\mathrm{5x}\), and we have \(\mathrm{5x = 12}\)
• No need to solve for x individually!

Answer: C. 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students solve the entire system to find x and y individually, missing that the question only asks for \(\mathrm{5x}\).

They continue: \(\mathrm{5x = 12}\), so \(\mathrm{x = \frac{12}{5} = 2.4}\), then \(\mathrm{y = 3(2.4) = 7.2}\). Then they look for \(\mathrm{x = 2.4}\) among the answer choices, don't find it, and end up guessing or selecting Choice D (5) thinking it relates to the coefficient.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when combining like terms.

They might incorrectly combine \(\mathrm{2x + 3x}\) as \(\mathrm{6x}\) instead of \(\mathrm{5x}\), leading to \(\mathrm{6x = 12}\), so \(\mathrm{x = 2}\). This might cause them to select Choice D (5) or get confused and guess.

The Bottom Line:

This problem rewards students who read carefully and recognize when they've found exactly what's being asked for. The key insight is stopping at \(\mathrm{5x = 12}\) rather than continuing to solve for individual variables.