1. **Stating the problem:** We are given the partial differential equation (PDE) $$(v - z) p + (x + y) q = z - x$$ where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$. 2. **Understanding the PDE:** This is a first-order linear PDE in terms of $z(x,y)$. 3. **Method of characteristics:** We use the method of characteristics to solve it. The characteristic equations are: $$\frac{dx}{v - z} = \frac{dy}{x + y} = \frac{dz}{z - x}$$ 4. **Solve the system:** From $$\frac{dx}{v - z} = \frac{dy}{x + y}$$ and $$\frac{dy}{x + y} = \frac{dz}{z - x}$$, we find two independent integrals (constants along characteristics). 5. **First characteristic equation:** $$\frac{dx}{v - z} = \frac{dy}{x + y}$$ Cross-multiplied: $$(v - z) dy = (x + y) dx$$ 6. **Second characteristic equation:** $$\frac{dy}{x + y} = \frac{dz}{z - x}$$ Cross-multiplied: $$(z - x) dy = (x + y) dz$$ 7. **Solve the first equation:** Rewrite as: $$\frac{dy}{dx} = \frac{x + y}{v - z}$$ But $v$ is a variable or parameter; since $v$ is not defined explicitly, we treat $v$ as a constant parameter. 8. **Solve the second equation:** Rewrite as: $$\frac{dy}{dz} = \frac{x + y}{z - x}$$ 9. **Summary:** The solution involves finding two independent functions $F$ and $G$ constant along characteristics, then the general solution is: $$F(\text{first integral}, \text{second integral}) = 0$$ 10. **Final answer:** The PDE solution is implicit in the characteristic variables defined by: $$\frac{dx}{v - z} = \frac{dy}{x + y} = \frac{dz}{z - x}$$ This defines the family of solutions to the PDE. **Note:** Without additional initial/boundary conditions or explicit $v$ definition, this is the general implicit solution form.