1/2y = 4x - 1/2y = 2The system of equations above has solution \((\mathrm{x}, \mathrm{y})\). What is the value of...

1. TRANSLATE the problem information

• Given system:
  • \(\frac{1}{2}\mathrm{y} = 4\)
  • \(\mathrm{x} - \frac{1}{2}\mathrm{y} = 2\)
• Find: The value of x

2. INFER the most efficient approach

• Notice that both equations contain the term \(\frac{1}{2}\mathrm{y}\)
• One equation has \(+\frac{1}{2}\mathrm{y}\), the other has \(-\frac{1}{2}\mathrm{y}\)
• Adding the equations will eliminate y completely, leaving only x

3. SIMPLIFY by adding the equations

• Add left sides: \((\frac{1}{2}\mathrm{y}) + (\mathrm{x} - \frac{1}{2}\mathrm{y}) = \mathrm{x}\)
• Add right sides: \(4 + 2 = 6\)
• Result: \(\mathrm{x} = 6\)

Answer: D. 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the elimination opportunity and instead try substitution method first.

They solve the first equation to get \(\mathrm{y} = 8\), then substitute into the second equation: \(\mathrm{x} - \frac{1}{2}(8) = 2\). While this works, they may make arithmetic errors like \(\mathrm{x} - 4 = 2\), leading to \(\mathrm{x} = -2\) instead of \(\mathrm{x} = 6\). This leads to confusion since -2 isn't among the choices, causing them to guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students attempt elimination but make sign errors when adding.

They might incorrectly add as: \(\frac{1}{2}\mathrm{y} + \mathrm{x} + \frac{1}{2}\mathrm{y} = 4 + 2\), getting \(\mathrm{x} + \mathrm{y} = 6\) instead of just \(\mathrm{x} = 6\). This creates confusion about what they've solved for, leading them to select Choice C (4) if they substitute back incorrectly.

The Bottom Line:

This problem rewards students who can quickly spot the elimination pattern. The key insight is recognizing that the coefficients of y are opposites, making addition the most direct path to the solution.