If algebra is a branch of mathematics dealing with finding the unknown values or quantities, then I can say something like $1+1=x$ is an algebraic equation.
This kind of problems for sure is not related to algebra, it's all about normal computation, but I could write it this way because there is no certain rule prevents me from doing it, right?.
Is there any kind of rigorous definition to algebra that Distinguishes it from this Vague?
Algebra are arithmetic are essentially the same thing, and I think it is a shame that most teachers in schools act as if adding letters into equations is a huge leap from simple computation. When we first learn math, we are given problems like $$\text{What is }9-5\text{?}$$ However, this question could be asked as $$9-5=\text{?}$$ Or, $$9-5=x$$ As long as students understand the idea that if we apply equal operations to equal things, they stay equal, then algebraic manipulations like $$x+5=9$$ $$x=9-5$$ $$x=4$$ should feel very natural.