1. **State the problem:** Solve the equation $$z = px + qy + \sqrt{1 + p^2 + q^2}$$ for variables or interpret its meaning. 2. **Understanding the equation:** This equation expresses $$z$$ as a function of $$x$$ and $$y$$ with parameters $$p$$ and $$q$$. The term $$\sqrt{1 + p^2 + q^2}$$ is a constant with respect to $$x$$ and $$y$$. 3. **Interpretation:** This is the equation of a plane in 3D space with slope components $$p$$ and $$q$$ in the $$x$$ and $$y$$ directions respectively, and a vertical offset $$\sqrt{1 + p^2 + q^2}$$. 4. **If solving for $$z$$:** The equation is already solved for $$z$$. 5. **If solving for $$p$$ or $$q$$ given $$z, x, y$$:** Rearrange the equation: $$z - \sqrt{1 + p^2 + q^2} = px + qy$$ This is nonlinear due to the square root term. 6. **Example:** If you want to isolate $$p$$ or $$q$$, more information or constraints are needed. 7. **Summary:** The equation defines $$z$$ in terms of $$x, y, p, q$$ with a nonlinear offset. Without additional conditions, it is already solved for $$z$$. **Final answer:** $$z = px + qy + \sqrt{1 + p^2 + q^2}$$