I think WolframAlpha has a limit on the number of characters that can make up any query.

Your problem as you have shown it takes 192 characters to enter into WolframAlpha and I think that might be almost twice the length that seems to be acceptable.

If I change your problem to be

y=mod(x,11),9812+x*z=11288,15780-x*y=11844

then it immediately responds with a solution x=492, y=8, z=3.

Perhaps you can use this as an example and see if you can get a solution to your actual problem.

Sometimes if a problem takes too many characters to write it might be possible to solve one simpler problem and then use that solution as input to solve a second problem which gives you the desired answer.

If as a second problem I give WolframAlpha

solve 492=5*13+6*12+c*11+d*10+e*9+f*8+6*7+5*6+i*5+j*4+7*3+9*2 over the integers

and I ask it to use i as a variable instead of the square root of -1 then it immediately responds that there is no solution over the integers.

Please check all this very carefully to make certain that I have not made any mistake and have not misunderstood your problem.

It seems unusual that you are multiplying one of your variables by 11 and also working modulo 11. That would seem to allow any value for that variable.

I am not sure your problem is described precisely enough for me to know I am trying to solve the correct thing.