The problem asks to use equilibrium equations to find support reactions. The equilibrium equations given are the sum of vertical forces equals zero ($\Sigma Fy = 0$) and the sum of moments equals zero ($\Sigma M = 0$). However, there is no specific structural system or loading given. Therefore, a general explanation of how to apply these principles will be provided, but a numerical solution cannot be given without a specific problem.

1. Problem Description

The problem asks to use equilibrium equations to find support reactions. The equilibrium equations given are the sum of vertical forces equals zero ($\Sigma Fy = 0$) and the sum of moments equals zero ($\Sigma M = 0$). However, there is no specific structural system or loading given. Therefore, a general explanation of how to apply these principles will be provided, but a numerical solution cannot be given without a specific problem.

2. Solution Steps

The fundamental principles of static equilibrium are:

* The sum of all forces acting on a body must be zero. This means the sum of the forces in the x-direction ($\Sigma Fx = 0$), the sum of the forces in the y-direction ($\Sigma Fy = 0$), and the sum of the forces in the z-direction ($\Sigma Fz = 0$) must each be zero.

* The sum of all moments acting on a body about any point must be zero ($\Sigma M = 0$). The choice of the point about which moments are summed is arbitrary, but a strategic choice can often simplify the calculations. Moments are calculated as force times distance. The sense of the moment (clockwise or counterclockwise) needs to be carefully considered, and a consistent sign convention must be used.

To find support reactions for a structure:

1. Draw a free body diagram (FBD) of the structure. This involves representing the structure with all external forces and support reactions. Different types of supports will have different types of reactions. For example, a pin support can have horizontal and vertical reactions, while a roller support typically has only a vertical reaction.

2. Apply the equilibrium equations. Start with $\Sigma Fy = 0$. This equation relates the vertical forces and support reactions. If there is only one unknown vertical reaction, it can be solved directly. If there are multiple unknowns, other equations must be used.

3. Next, apply $\Sigma M = 0$. Choose a point to sum moments about that will eliminate one or more unknown reactions from the equation. This is often done by choosing a point at a support where the reaction is unknown, because the moment arm for the reaction at that point will be zero. This leaves fewer unknowns in the moment equation. Solve the moment equation for the remaining unknown reaction.

4. Finally, use the remaining equilibrium equations (including $\Sigma Fy = 0$ if not already used, and possibly $\Sigma Fx = 0$ if there are horizontal forces or reactions) to solve for the remaining unknown support reactions.

5. Check the solution by verifying that all equilibrium equations are satisfied with the calculated reaction values.

3. Final Answer

Without a specific problem statement with a structural system and loading, the final answer cannot be a numerical value for the support reactions. The process to find support reactions involves drawing a free body diagram, applying $\Sigma Fy = 0$, and $\Sigma M = 0$ to solve for the unknown reactions.