I'm trying to justify the following simplification: $(x + y) + (y \cdot x') = x + y$
The solution that was provided to me is as follows:
$ = (x + y) + (y \cdot x') - (LHS)\\ = x + y \cdot (1 + x')\\ = x + y \cdot (1)\\ = x + y\\ $
I'm a little confused on the jump from steps 1 to 2, and what rule was applied and then also the switch from a disjunction to a conjunction.
Is anyone able to provide some insight into the reasoning?
That step actually combines three steps. In order, they are associativity of disjunction, conjunctive identity, and distributivity of conjunction over disjunction:
$$\begin{align*} (x+y)+(y\cdot x')&=x+\big(y+(y\cdot x')\big)\\ &=x+\big((y\cdot 1)+(y\cdot x')\big)\\ &=x+\big(y\cdot(1+x')\big) \end{align*}$$
Then it just completes the calculation within the parentheses.