Let (a-1) d = b (c-1) such that $ a,b,c,d \in \mathbb{R} $.
How do you find the relationships between a,b,c,d? Do you look at what makes both sides equivalent?
I considered 3 cases where:
- a = 1, b = 0 and $c,d \in \mathbb{R} $
- $ (a-1) \neq 0 $, $ b \neq 0 $, c = 1, and $d \in \mathbb{R} $
- $ (a-1) \neq 0 $, $\mspace{4mu}b,\mspace{4mu}c \in \mathbb{R} $ and $ d =b(c-1)/(a-1) $
Should I be considering other cases such as isolating b in the same way as isolating d or would that be redundant?
i would write: first case: $$d=0$$ then we have $$0=b(c-1)$$ and we get $$b=0$$ or $$c=1$$. Now we consider the case $$d\neq 0$$ and we can write $$a=\frac{b}{d}(c-1)+1$$