If you have an equation like 2x0
, it is perfectly clear that the only possible solution is x->0
. However, if you have an equation like ax0
, things are not so clear. If a
is not equal to zero, then x->0
is again the only solution. However, if a
is in fact equal to zero, then any
value of x
is a solution. You can see this by using Reduce
A basic difference between Reduce
is that Reduce
the possible solutions to a set of equations, while Solve
gives only the generic
ones. Solutions are considered "generic" if they involve conditions only on the variables that you explicitly solve for, and not on other parameters in the equations. Reduce
also differ in that Reduce
always returns combinations of equations, while Solve
gives results in the form of transformation rules.
When you have several simultaneous equations, Reduce
can show you under what conditions the equations have solutions. Solve
shows you whether there are any generic solutions.
When you work with systems of linear equations, you can use Solve
to get generic solutions, and Reduce
to find out for what values of parameters solutions exist.
For nonlinear equations, the conditions for the existence of solutions can be much more complicated.