a student solved an equation and found that the variable was eliminated in the process of solving the equation. how would the student know whether the equation is an identity or an equation with no solution?

Well if it was an identity you would be able to insert any number in the equation and the equation would still be true.  If it had no solution, it would just look like two integers that equal each other

For example:
No solution
8 = 0

Sometimes life is like a tunnel. you can’t always see the light at the end of the tunnel, but if you keep moving, you will come to a better place