The words al-Khwarizmi used -- al-jabr and al-muqabala -- actually meant "restoring" and "balancing", and referred to adding or subtracting some quantity from both sides of an equation. Muslim mathematicians of the day did not deal with negatives, so today we would consider al-jabr and al-muqabala to be variants of the same operation.

Nevertheless, the original writings were what suggested to me that the important bit of algebra was doing things to equations to change them into some easily solvable canonical form (al-Khwarizmi's work dealt primarily with quadratics).

Thanks for this, I blended al-khwarizmi's book title with "Kitāb al-Jamʿ wa-l-tafrīq bi-ḥisāb al-Hind (The Book of Bringing_together and Separating According to the Hindu Calculation)" mentioned in its wikipedia page.