combining like terms

If you have three boxes of nails and then bring in four more boxes, you simply say seven boxes — you count the boxes, not re-describe each nail. Combining like terms works the same way: you add or subtract the counts (the coefficients) of terms that are the same kind, and keep the variable part unchanged.

The rule rests on the distributive property: 3x + 4x is the same as (3 + 4)x = 7x, because x is a common factor. You only ever add the coefficients; the variable part rides along untouched. So 5a - 2a = 3a, and 6x^2 + x^2 = 7x^2 (here the unwritten coefficient of x^2 is 1). You may never combine unlike terms — 3x + 4y stays as 3x + 4y, with nothing to merge.

Watch the signs carefully: each term carries the sign written in front of it. In 7x - 10x the coefficients are +7 and -10, so the result is -3x, not 3x. Pulling like terms together is usually the very first move when you simplify an expression.