1. **State the problem:** Given the equation $$2x = \frac{2y^2 - 3z}{2}$$, express $$y$$ in terms of $$x$$ and $$z$$. 2. **Write down the formula and rules:** We want to isolate $$y$$. The equation involves a fraction and a quadratic term $$y^2$$. 3. **Multiply both sides by 2 to eliminate the denominator:** $$2 \times 2x = 2y^2 - 3z$$ $$\Rightarrow 4x = 2y^2 - 3z$$ 4. **Add $$3z$$ to both sides:** $$4x + 3z = 2y^2$$ 5. **Divide both sides by 2 to isolate $$y^2$$:** $$y^2 = \frac{4x + 3z}{2}$$ 6. **Take the square root of both sides to solve for $$y$$:** $$y = \pm \sqrt{\frac{4x + 3z}{2}}$$ **Final answer:** $$y = \pm \sqrt{\frac{4x + 3z}{2}}$$