Once you can write an expression in algebra, you can often make it shorter without changing its value. The trick is to collect like terms — terms built from the same letters — into one.

Think of 3a as three a-tiles and 2a as two a-tiles. Put them in one pile and you have five a-tiles:

Nothing magic happened — it is the same as 3 + 2 = 5, just counting a's instead of plain numbers. This is the distributive law read backwards: 3a + 2a = (3 + 2)a = 5a.

The picture below makes it visual. Tiles of the same kind slide together into one group; tiles of different kinds stay apart. Step through it.

Unlike terms stay apart

Terms only combine when their letters match exactly. You cannot add an x to a y, and you cannot add an x to an x^2 — they are different kinds of tile. So in

the two x terms join (2x + x = 3x), but the 3y has no partner and is left as it is. A simplified expression keeps one term of each kind.

See it explained

Khan Academy introduces combining like terms here: