Show that the class of languages generated by right linear grammars is exactly the class of regular languages. Two part problem.

Part $1.$ Given a right$-$linear grammar G$,$ show how to build an equivalent NFA M$.$ Then, show every string in L$($G$)$ is also in L$($M$),$ and every string in L$($M$)$ is also in L$($G$).$

Part $2.$ Given a DFA M$,$ show how to build an equivalent right$-$linear grammar G$.$ Then, show every string in L$($G$)$ is also in L$($M$),$ and every string in L$($M$)$ is also in L$($G$)$