6x + 7y = 28 2x + 2y = 10 The solution to the given system of equations is \((\mathrm{x},...

1. TRANSLATE the problem information

• Given system:
  • \(6\mathrm{x} + 7\mathrm{y} = 28\)
  • \(2\mathrm{x} + 2\mathrm{y} = 10\)
• Find: the value of y

2. INFER the solution strategy

• Both equations have x and y terms, so we need elimination or substitution
• Notice that if we multiply the second equation by 3, we get \(6\mathrm{x} + 6\mathrm{y} = 30\)
• This gives us matching x-coefficients (both 6x), perfect for elimination

3. SIMPLIFY using elimination method

• Multiply second equation by 3:
  \(3(2\mathrm{x} + 2\mathrm{y}) = 3(10)\)
  \(6\mathrm{x} + 6\mathrm{y} = 30\)
• Now we have:
  • \(6\mathrm{x} + 7\mathrm{y} = 28\)
  • \(6\mathrm{x} + 6\mathrm{y} = 30\)
• Subtract the second from the first:
  \((6\mathrm{x} + 7\mathrm{y}) - (6\mathrm{x} + 6\mathrm{y}) = 28 - 30\)

4. SIMPLIFY to find y

• Distribute the subtraction:
  \(6\mathrm{x} + 7\mathrm{y} - 6\mathrm{x} - 6\mathrm{y} = -2\)
• Combine like terms:
  \((6\mathrm{x} - 6\mathrm{x}) + (7\mathrm{y} - 6\mathrm{y}) = -2\)
• Simplify:
  \(0 + \mathrm{y} = -2\)
• Therefore:
  \(\mathrm{y} = -2\)

Answer: A. -2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students solve the system correctly but answer with the value of x instead of y

After finding \(\mathrm{y} = -2\), they substitute back to get \(\mathrm{x} = 7\), but then mistakenly select x as their final answer since 7 appears as an answer choice. This may lead them to select Choice B (7).

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors during the elimination process

Common mistakes include: wrong signs when subtracting equations, incorrect multiplication of the second equation, or errors when combining like terms. These calculation errors lead to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests both systematic equation-solving skills and careful attention to what the question is asking for. Success requires methodical elimination technique combined with reading comprehension to identify the correct variable.