Derive Algebraic Expressions. Parallel and Series Resistors.

1. Four resistors R₁, R₂, R₃, and R₄ are connected in parallel. A drop of the electric potential on the resistor R₁ is V₁. Derive an algebraic expression for the drop of the electric potential on the resistor R₄ in the terms of R₁, R₂, R₃, R₄, and V₁.

2. Four resistors R₁, R₂, R₃, and R₄ are connected in series. An electric current through the resistor R₁ is I₁. Derive an algebraic expression for the electric current through the resistor R₄ in the terms of R₁, R₂, R₃, R₄, and I₁.

3. Four resistors R₁, R₂, R₃, and R₄ are connected in series. A drop of the electric potential on the resistor R₁ is V₁. Derive an algebraic expression for the drop of the electric potential on the resistor R₄ in the terms of R₁, R₂, R₃, R₄, and V₁.

4. Four resistors R₁, R₂, R₃, and R₄ are connected in parallel. An electric current through the resistor R₁ is I₁. Derive an algebraic expression for the electric current through the resistor R₄ in the terms of R₁, R₂, R₃, R₄, and I₁.

5. Four resistors with known resistances R₁, R₂, R₃, and unknown R₄ are connected in series. An equivalent resistance is R. Derive an algebraic expression for the unknown resistance R₄ in the terms of known parameters. 

6. Four resistors with known resistances R₁, R₂, R₃, and unknown R₄ are connected in parallel. An equivalent resistance is R. Derive an algebraic expression for the unknown resistance R₄ in the terms of known parameters. 

7. Four resistors with known resistances R₁, R₂, R₃, and unknown R₄ are connected in series to a battery with a known terminal voltage V. The total power P of electric energy being dissipated in all resistors is known. Derive an algebraic expression for the unknown resistance R₄ in the terms of known parameters. 

8. Four resistors with known resistances R₁, R₂, R₃, and unknown R₄ are connected in parallel to a battery with a known terminal voltage V. The total power P of electric energy being dissipated in all resistors is known. Derive an algebraic expression for the unknown resistance R₄ in the terms of known parameters.