I am aware that there is another person who can tell you about equality sign $=$ in more rigor and axiomatic way, but the main idea is that if $x=y$ for two abstract elements $x,y$ then they are the same.
Further, by axioms of a linear space $V$ and space of real numbers $\mathbb R$, for each $x,y\in V$ and $\alpha,\beta\in\mathbb R$ there exists the unique element $z\in V\text{ s.t. } z = \alpha x+\beta y$. That means that if x'=x'' then x'+y = x''+y for any $y$ since the result of addition is unique and hence both sides are equal.
Now, suppose you have two equalities: $x=a$ and $y=b$. $ x=a\Rightarrow x+y = a+y\Rightarrow x+y = a+b $ and that is how you add equations. The subtraction works in the same way.
How do you apply it for the equations. Suppose you have two equations for two unknowns $s$ and $t$: $ \begin{cases} x(s,t) = a, \\ y(s,t) = b.\quad(1) \end{cases} $
You know that if $s,t$ satisfy $(1)$ then $x(s,t) = a$ and $y(s,t) = b$ so $x(s,t)+y(s,t) = a+b$.