What's a solution with a variable that has no solution

i think you meant to say "whats an equation with a variable that has no solution". if so, then this would be considered a contradiction.

consider the following example

x+3 = x+5

if we subtract x from both sides, they will cancel out and go away

x+3-x = x+5-x

x-x+3 = x-x+5

0x+3 = 0x+5

0+3 = 0+5

3 = 5 this is where the contradiction happens

the last equation above is false regardless of what x value you pick; therefore, there are no solutions to the example equation.