"but rarely example how to use maxima
as a simple helper doing manual calculations."
You can also add, subtract, multiply and divide both sides of an equation by a variable, constant or another equation like in %i8 and %i9 which can be a useful check when doing algebra.
(%i6) x + y = 9;
(out6) y + x = 9
(%i7) x + 3*y = 5;
(out7) 3 y + x = 5
(%i8) %-x;
(out8) 3 y = 5 - x
(%i9) out6-x;
(out9) y = 9 - x
(%i10) subst(out9, out8);
(out10) 3 (9 - x) = 5 - x
(%i11) first(solve(%,x));
(out11) x = 11
(%i12)