How to multiply an equation by a factor

Questions
1

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
2

1/a* (ax + by + c*z)==0//Expand

3

Try this code

eq = ax + by + c*z == 0;
eqSplit = Apply[List, eq] ;
Apply[Equal, (1/a)*eqSplit // Expand]

4

eqnlist/.{a_= =b_ :> (Simplify/@(a x == b x)}

or if cancellations of factors are intended

eqnlist/. {a_== b_ :> (0==#&) /@ Factor[ Simplify[a-b]}]

And finally in order to get the logical solution set of equations

Or @@ (Solve[{a*x + b*y + c*z == 0, 1/a*(a*x + b*y + c*z) == 0,    x + b/a*y + c/a*z == 0}] /. Rule -> Equal)


 {z == (-a x - b y)/c} || 
 {c == 0, y == -((a x)/b)} || 
 {b == 0, c == 0,   x == 0}
5

Take a look at MultiplySides and DivideSides.