Explanation below. No short answer available.
Graphing two lines to find their common root, if any, has its evident advantages, and its disadvantages. To find the root, you simply look at the point of intersection of the lines, if any. At first, this seems like a way easier method, but here is a table to help compare the methods:
Algebraic analysis means:
++Mathematical rigor, and accuracy
++Intuition for further use, and solving of systems of equation
+Easier understood by machines
-Sometimes hard to understand
+++Helps with geometry-based proofs, and has the best intuition
++Works for any function, not just lines.
-Graphing to scale can be difficult
--Gives a sense of inaccuracy, especially within the realms of calculus
Overall, I prefer the algebraic method, just because it's my kind of "universal pass". It works well, even though it can be hard at times.
However, graphing, especially when learning the concept, is definitely not a bad method, even if it seems to have more cons in my comparison table. That is because past a point, graphing doesn't help as much, until you learn the full analysis process, with the asymptotes and everything, allowing you to make a proper graph of more complex functions, which means that for the time being, the algebraic method might as well prove to be superior.