B. \( \left\{\begin{array}{l}x-y=2 \\ y=\frac{1}{2} x\end{array}\right. \)

To solve the system of equations, let's substitute \(y = \frac{1}{2} x\) into the first equation \(x - y = 2\). Replacing \(y\) gives us: \[ x - \frac{1}{2} x = 2 \] This simplifies to: \[ \frac{1}{2} x = 2 \] Multiplying both sides by 2, we find \[ x = 4. \] Now, substituting \(x\) back into the expression for \(y\): \[ y = \frac{1}{2} (4) = 2. \] So the solution to the system is \(x = 4\) and \(y = 2\). Now let’s also contemplate the graphical representation! When plotted, the lines represented by the equations intersect at the point (4, 2). This means that both conditions are satisfied at that specific point, showcasing how sometimes two different equations can share the same solution in the coordinate plane—it's like finding common ground in a conversation!