Inverse Operations
Inverse operations are pairs of mathematical operations that undo each other, such as addition and subtraction, or multiplication and division.
Definition
Inverse operations are pairs of operations that cancel each other out and restore an original value: addition and subtraction are inverse operations, and so are multiplication and division, along with squaring and square root, and exponentiation and logarithm. Using an inverse operation on both sides of an equation keeps it balanced, since each pair corresponds to a pair of group operations (or ring operations); addition and its inverse subtraction, for instance, define an abelian group structure on the integers. In group theory, every element $a$ has an inverse $a^{-1}$ such that $a \cdot a^{-1} = e$ (the identity), and solving an equation by applying inverse operations in reverse order is essentially composing the inverse function: if $f(x) = ax + b$, then $f^{-1}(y) = (y - b)/a$. Inverse operations generalize further to inverse functions and inverse matrices in higher mathematics.
Example
If you add $5$ to a number and then subtract $5$, you are back where you started; in an equation, if a variable has $5$ added to it, you subtract $5$ from both sides to undo it. To solve $4x = 28$, divide both sides by $4$ (the inverse of multiplying by $4$) to get $x = 7$; to solve $x^2 = 25$, take the square root of both sides to get $x = \pm 5$. Solving the matrix equation $Ax = b$ uses the matrix inverse, $x = A^{-1}b$, provided $A$ is invertible ($\det(A) \neq 0$), generalizing "dividing both sides" to the matrix setting.
Key Insight
Inverse operations are your tools for solving equations: each one "undoes" a step that was applied to the variable. The existence of inverses is the key property that distinguishes a group from a semigroup, and it is precisely what allows equations to be "solved" rather than merely approximated.