No, there is not always a solution to every equation. Infinitely many solutions is when both sides have the same equation. Ex: 4x - 1 = 4x - 1 No matter what you plug in.. it will be equal on both sides.
No solution is something like this: 4x=4x+1If you subtract the 4x, you get 0 = -1
That is obviously not equal to -1, so there is no solution.
No matter what number you plug in for x, it will NOT be equal on both sides.
To put it shortly:
Infinitely many solutions- EVERYTHING works
No solution- NOTHING works