Algebraically means 'by using algebra' or 'in the language of algebra' — that mathematics of letters and signs for numbers. If you solve a problem algebraically, you keep the letters and follow the rules of equations, rather than guessing a number or only drawing a graph. Two expressions can be algebraically the same: they look different at first, but they name the same thing once you tidy the letters. It is an adverb, so it likes to sit next to verbs such as solve, express, and simplify.