Solving a linear equation in Mathematica

Question

This should be easy but I can't seem to find the right way to do it.

I have an equation of the form $a x + b x + c y + a z + d z = 0$, and I'd like to solve for relations between the parameters $a,b,c,d$, etc. such that the equation holds for all values of $x,y,z$, etc.

This is a very simplified example of the kind of equation I'm dealing with; there are actually 308 variables $x,y,\ldots$ and 42 parameters $a,b,\ldots$, making this nontrivial. As in the short sample equation, different variables could have the same parameter and could appear multiple times with different parameters.

So far my best attempt is:

Solve[ForAll[{x, y, ...}, a*x + b*y + ... == 0], {a, b, ...}]

This yields the correct solution on a smaller equation (about 20 variables and 9 parameters) but is far too slow for this one. Is there a more specific technique I could use that's more optimized for this kind of problem specifically? Leaving out ForAll yields solutions in terms of the x-variables, which is kind of useless.

EDIT: nikie mentioned a much better way to do it (in the comments)! Thanks!

Answer 1

Coefficient[a x + b x + c y + a z + d z, #] == 0 & /@ {x, y, z} // Reduce[#, {a, b, c, d}] &

b == -a && c == 0 && d == -a

Answer 2

You can use SolveAlways for the type of problems you describe. I haven't looked if it's fast enough for a very large system though.

In[1]:= SolveAlways[a x + b x + c y + a z + d z == 0, {x, y, z}]

Out[1]= {{a -> -d, b -> d, c -> 0}}