The solution to the given system of equations is \(\mathrm{(x, y)}\). What is the value of x?y = -3x4x +...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{y = -3x}\) (first equation)
  • \(\mathrm{4x + y = 15}\) (second equation)
• Find: the value of x

2. INFER the solution strategy

• Since the first equation already expresses y in terms of x, substitution method is ideal
• Substitute the expression for y from the first equation into the second equation

3. SIMPLIFY through substitution and algebra

• Substitute \(\mathrm{y = -3x}\) into \(\mathrm{4x + y = 15}\):
  \(\mathrm{4x + (-3x) = 15}\)
• Combine like terms:
  \(\mathrm{4x - 3x = x = 15}\)
• Therefore: \(\mathrm{x = 15}\)

Answer: C. 15

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors or incorrectly combine like terms when working with \(\mathrm{4x + (-3x)}\). They might calculate this as \(\mathrm{4x + 3x = 7x}\) or forget that \(\mathrm{4x - 3x = x}\) (not 1x), leading to wrong intermediate steps and ultimately incorrect final answers.

This may lead them to select Choice A (1) or Choice B (5) depending on their specific arithmetic mistake.

Second Most Common Error:

Poor INFER reasoning about the solution target: Students solve the system correctly but find the value of y instead of x, or they find \(\mathrm{y = -45}\) and then select the absolute value. Since \(\mathrm{y = -3(15) = -45}\), they might see 45 as an answer choice and select it.

This may lead them to select Choice D (45).

The Bottom Line:

This problem tests whether students can execute the substitution method cleanly while keeping track of what variable they're solving for. The algebraic manipulation is straightforward, but small errors in signs or combining terms can derail the entire solution.