The problem is a structural analysis of a truss, requiring the computation of member forces and reactions at each joint. The setup asks for equilibrium equations to be written for the truss system shown, which is statically determinate and subjected to given loads.

Here's a breakdown of how to proceed:

1. Equilibrium at Joints C, D, and E:

   • We'll write equilibrium equations (force summations in both horizontal and vertical directions) for each joint.
   • Use the known relationships provided for forces at joints A and B and the geometric constraints of the truss to help.

2. Assembly into a Matrix:

   • After writing the equilibrium equations, organize them into matrix form to solve the system for the axial forces.

3. Reaction Forces:

   • Compute the reaction forces at nodes A and E after determining the axial forces.

4. Validation:

   • Verify consistency by checking the equilibrium at joint B and node E.

Let’s begin with equilibrium equations for each joint:

Equilibrium at Joint C:

• In the x-direction: $F_{4x} + F_{6x} = 0$
• In the y-direction: $F_{6y} = 10 \, \text{kN} \quad (\text{external load at C})$

Equilibrium at Joint D:

• In the x-direction: $F_{5x} + F_{6x} = 0$
• In the y-direction: $F_{5y} + F_{6y} = 0$

Equilibrium at Joint E:

• In the x-direction: $R_{Ex} + F_{6x} = 0$
• In the y-direction: $R_{Ey} + F_{6y} = 0$

The next step is to assemble these equations into a matrix form and solve for the axial forces $F_4, F_5, F_6$ as well as the reaction forces.

Would you like me to proceed with solving the matrix, or do you need further details on how to construct the equilibrium equations for other joints? Here are a few related questions:

1. What is the importance of pin joints in trusses for determining internal forces?
2. How do you verify that a truss is statically determinate?
3. What role does the geometric configuration (angles and lengths) play in the analysis of trusses?
4. How are support reactions calculated in a cantilever truss like this one?
5. Why are axial forces in trusses considered either tensile or compressive?

Tip: When solving for forces in trusses, breaking down components into x and y directions and using symmetry can simplify complex systems.