The horizontal beam ABC is loaded by a distributed load of intensity $5k\frac{N}{m}.$ The beam is supported at A with a roller which runs on an inclined plane and it is hinged to the vertical bar at C and hinged to the link BD$.$ Knowing that the joints at D and E are hinges then:

a$)$ Determine the reactions at supports A and E $.$

b$)$ Derive the equations that describe the variation of normal, shear and bending moment along beam ABC $.$

c$)$ Draw the normal, shear and bending moment diagrams for the beam ABC $.$