Explain the 2 different answers you can get when solving an equation and the variable is dropped out

To give a simple answer

if you consider the equation to be that of a line.
then if the variables cancel out, then the two lines will have the same slope.

two lines with the same slope fall under two categories. Parallel lines, and lines on top of each other

parallel lines have no solutions since the lines never cross each other
lines on top of each other have infinitely many solutions  since they "cross" everywhere on the line