How do you know when an equation only has one solution, no solution, and infinite.

I believe that this belongs in the math genre, but I'll answer it anyway.

Say we have the equation "4x-3= 9-x," you would then plug in a number for the variable. I'll choose the number, 1. Now the equation looks like "4(1)-3=9-1," correct? Now we solve it from there. "4-3=9-1," is also equal to "1=8." Now is this correct? No, which means it has no solutions. Unless anther number works, in that case it becomes one solution. To determine whether it's infinite or only one, you can plug in more numbers. If you plug in a different number and it does work, that equation has infinite solutions. If it does not work with any number besides one, it only has one solution.

Sorry that this is so drawn out, I hope I helped!