1. **Problem statement:** Given functions $$u = \frac{2x - y}{2} x$$ and $$v = \frac{y}{2},$$ find the Jacobian determinant \(\frac{\partial(u,v)}{\partial(x,y)}\). 2. **Rewrite the functions:** $$u = \frac{2x - y}{2} x = x \left( \frac{2x - y}{2} \right) = \frac{2x^2 - xy}{2},$$ $$v = \frac{y}{2}.$$ 3. **Calculate the partial derivatives:** $$\frac{\partial u}{\partial x} = \frac{\partial}{\partial x} \left( \frac{2x^2 - xy}{2} \right) = \frac{\partial}{\partial x} \left( x^2 - \frac{xy}{2} \right) = 2x - \frac{y}{2},$$ $$\frac{\partial u}{\partial y} = \frac{\partial}{\partial y} \left( \frac{2x^2 - xy}{2} \right) = \frac{\partial}{\partial y} \left( x^2 - \frac{xy}{2} \right) = -\frac{x}{2},$$ $$\frac{\partial v}{\partial x} = \frac{\partial}{\partial x} \left( \frac{y}{2} \right) = 0,$$ $$\frac{\partial v}{\partial y} = \frac{\partial}{\partial y} \left( \frac{y}{2} \right) = \frac{1}{2}.$$ 4. **Write the Jacobian matrix:** $$J = \begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{pmatrix} = \begin{pmatrix} 2x - \frac{y}{2} & -\frac{x}{2} \\ 0 & \frac{1}{2} \end{pmatrix}.$$ 5. **Calculate the Jacobian determinant:** $$\frac{\partial(u,v)}{\partial(x,y)} = \left( 2x - \frac{y}{2} \right) \cdot \frac{1}{2} - \left( -\frac{x}{2} \right) \cdot 0 = \frac{2x - \frac{y}{2}}{2} = x - \frac{y}{4}.$$ **Final answer:** $$\boxed{\frac{\partial(u,v)}{\partial(x,y)} = x - \frac{y}{4}}.$$