מערכת יחסים is such a long word...can anyone explain the smaller pieces?
מערכת = set/system
יחסים = relations
Yesh lanu ma'arechet yachasim tova.
Does מערכת יחסים mean a business partnership, a love relationship, or both?
It could be either.
Does the word יחסים have anything to do with simcha (a celebration)? Thank you.
It's a different root; י.ח.ס vs ש.מ.ח. If the similarity of sounds helps you remember it, then great! But the root is not the same.
if you change the מערכת to מערכה - would that mean a bad relationship?
No. The word מערכה has various meanings, but none of them would be translated so.
Besides, in contrast state מערכה becomes מערכת, so in fact מערכת יחסים could be מערכה+יחסים (but it's not).