Eg.
a*x + b*y + c*z==0, 1/a* (a*x + b*y + c*z)==0, x + b/a*y + c/a*z==0
Take a look at MultiplySides and DivideSides.
MultiplySides
DivideSides
eq = ax + by + c*z == 0; eqSplit = Apply[List, eq] ; Apply[Equal, (1/a)*eqSplit // Expand]
1/a* (a*x + b*y + c*z)==0//Expand
