**Simultaneous Equations in Slope Deflection Method**

In the slope deflection method, simultaneous equations are derived to solve for the unknown member end moments and rotations in a structural system. These equations are obtained by applying the equilibrium and compatibility conditions at each joint of the structure.

The equilibrium conditions state that the sum of the moments at each joint must be equal to zero. The compatibility conditions ensure that the rotations and displacements at each joint are consistent with the applied loads and the deformation of the structure.

**Methods for Solving Simultaneous Equations:**

1. **Moment Distribution Method:**
The moment distribution method is an iterative technique used to solve the simultaneous equations in the slope deflection method. It involves the distribution of moments at the joints based on the stiffness of the members. The process is repeated until the moments converge to a stable solution.

2. **Williot Mohr Method:**
The Williot Mohr method is another approach for solving the simultaneous equations in the slope deflection method. It involves the transformation of the structural system into a statically determinate beam using the principles of reciprocal theorem. This method simplifies the solution of the simultaneous equations by eliminating the need for iteration.

3. **Consistent Deformation Method:**
The consistent deformation method is a systematic approach for solving the simultaneous equations in the slope deflection method. It involves the calculation of member end moments and rotations based on a consistent deformation of the structure. The method ensures that the equilibrium and compatibility conditions are satisfied simultaneously.

4. **Conjugate Beam Method:**
The conjugate beam method is an alternative technique for solving the simultaneous equations in the slope deflection method. It involves the transformation of the structural system into an equivalent conjugate beam, where the moments and rotations are related through a conjugate relationship. This method provides a convenient way to solve the simultaneous equations without the need for iteration.

**Conclusion:**
The simultaneous equations in the slope deflection method can be solved by various iterative and non-iterative techniques such as the moment distribution method, Williot Mohr method, consistent deformation method, and conjugate beam method. Each method has its advantages and can be applied depending on the complexity of the structural system and the desired level of accuracy. It is important to understand and apply the appropriate method to obtain accurate and reliable results in the analysis and design of structures.