1. **State the problem:** Make $x$ the subject of the formula given by $$6x - 2y = 2xz + 5u.$$\n\n2. **Rewrite the equation:** We want to isolate $x$ on one side. The equation is $$6x - 2y = 2xz + 5u.$$\n\n3. **Bring all terms involving $x$ to one side:** Subtract $2xz$ from both sides to get $$6x - 2xz - 2y = 5u.$$\n\n4. **Factor out $x$ from terms on the left:** $$x(6 - 2z) - 2y = 5u.$$\n\n5. **Add $2y$ to both sides:** $$x(6 - 2z) = 5u + 2y.$$\n\n6. **Divide both sides by $(6 - 2z)$ to isolate $x$:** $$x = \frac{5u + 2y}{6 - 2z}.$$\n\n7. **Simplify the denominator if possible:** Factor out 2: $$6 - 2z = 2(3 - z).$$ So, $$x = \frac{5u + 2y}{2(3 - z)}.$$\n\n**Final answer:** $$x = \frac{5u + 2y}{2(3 - z)}.$$