Algebraic Transformations of Equations

The most common transformations of algebra applied to equalities (the most common case) are summarized below, with with examples. In all of the subsections below, we assume that we are operating on equations expressed in terms of a set of variables with symbols such as $a, b, c, \ldots x, y, z$. All the rules and transformation below will work for complex numbers, real numbers (as a subset of complex numbers), or integers (as a subset of real numbers). They will not all necessarily work for more complicated algebras where the variables can stand for e.g. matrices or quaternions.

We begin by discussing equality itself. Equality is a somewhat subtle quality in algebra.