The final answer is important, but the steps are in some ways far more important.
I do many long computations using algebra and more fancy types of mathematics (calculus, complex analysis, differential geometry etc).
Some of the computations take hundreds of pages. Keeping track of the manipulations, being able to check the work for errors (which inevitably creep in), being able to redo the work later for the same or a related problem, being able to check the assumptions and so on are absolutely critical. And to do this, one has to write out all the steps, as clearly and carefully as possible.
Showing the steps is of vital importance if you want to do this kind of work. If you want to be a scientist, or an engineer, or an accountant, or statistician, or any kind of technical person, or to work in any profession where you have to interact with technical people (and these are some of the highest paying, most interesting jobs that exist), you have to be able to analyze complicated problems, and describe to yourself and others carefully how you solved them.
Even if you do not want to work in a profession involved with technical work, the entire reason you are studying algebra is to train yourself to think logically about certain kinds of problems. By not showing your steps, you are shortchanging yourself and hurting yourself, because showing your steps forces you to understand the process of getting an answer much better.
A final reason to show your steps is that if you show your work, it is far less likely that you will be accused of cheating. It is just smart to protect yourself from any kinds of charges like this, if you can.