nLab solution

A solution to an equation $f(x) = g(x)$ is a choice of value $v$ for the free variable $x$ such that the statement $f(v) = g(v)$ becomes true.

If we think of the equation as embodied by a pair of parallel morphisms

$$X_1 \underoverset{\underset{f_2}{\longrightarrow}}{\overset{f_2}{\longrightarrow}}{\phantom{AAAA}} X_2$$

then a solution is a cone over this diagram. The solution space itself is the limit cone, hence the equalizer of the diagram.