Let's see an example of three PDEs:

How can I eliminate variables q1[x,y,z] and q2[x,y,z] from system of three PDEs. In final form I need just one equation in function of p1[x,y,z].
$$\text{a1}=\text{A1}\frac{\partial ^2\text{p1}(x,y,z)}{\partial z^2}-\text{A2} \left(\frac{\partial ^2\text{p1}(x,y,z)}{\partial x^2}+\frac{\partial ^2\text{p1}(x,y,z)}{\partial y^2}+\frac{\partial\text{q1}(x,y,z)}{\partial x^1}+\frac{\partial \text{q2}(x,y,z)}{\partial y^1}\right)=0;$$

$$\text{a2}=\text{A9}\left(\text{A2} \frac{\partial }{\partial x^1}\left(\frac{\partial \text{q1}(x,y,z)}{\partial x^1}+\frac{\partial \text{q2}(x,y,z)}{\partial y^1}\right)+\text{A2} \left(\frac{\partial ^2\text{q1}(x,y,z)}{\partial x^2}+\frac{\partial ^2\text{q1}(x,y,z)}{\partial y^2}\right)\right)-\text{A4} \left(\frac{\partial \text{p1}(x,y,z)}{\partial x^1}+\text{q1}(x,y,z)\right)-\text{A8} \frac{\partial ^2\text{q1}(x,y,z)}{\partial z^2}=0; $$

$$\text{a3}=-\text{A1} \left(\frac{\partial \text{p1}(x,y,z)}{\partial y^1}+\text{q2}(x,y,z)\right)-\text{A4} \frac{\partial ^2\text{q2}(x,y,z)}{\partial z^2}+\text{A9} \left(\text{A6} \frac{\partial }{\partial y^1}\left(\frac{\partial \text{q1}(x,y,z)}{\partial x^1}+\frac{\partial \text{q1}(x,y,z)}{\partial y^1}\right)+\text{A7} \left(\frac{\partial ^2\text{q1}(x,y,z)}{\partial y^2}+\frac{\partial ^2\text{q2}(x,y,z)}{\partial x^2}\right)\right)=0;$$