1. **State the problem:** We have a set $A$ of all lines in the $xy$-plane and a relation $R$ on $A$ defined by $R = \{(L_1, L_2) : L_1 \parallel L_2\}$. We need to show that $R$ is an equivalence relation and find all lines related to the line $y = 3x + 5$. 2. **Recall the definition of equivalence relation:** A relation $R$ on a set is an equivalence relation if it is reflexive, symmetric, and transitive. 3. **Show reflexivity:** For any line $L$, it is parallel to itself because every line is parallel to itself by definition. So, $(L, L) \in R$. Thus, $R$ is reflexive. 4. **Show symmetry:** If $(L_1, L_2) \in R$, then $L_1 \parallel L_2$. Since parallelism is symmetric, $L_2 \parallel L_1$, so $(L_2, L_1) \in R$. Thus, $R$ is symmetric. 5. **Show transitivity:** If $(L_1, L_2) \in R$ and $(L_2, L_3) \in R$, then $L_1 \parallel L_2$ and $L_2 \parallel L_3$. Since if two lines are parallel to a third line, they are parallel to each other, $L_1 \parallel L_3$, so $(L_1, L_3) \in R$. Thus, $R$ is transitive. 6. **Conclusion:** $R$ is reflexive, symmetric, and transitive, so $R$ is an equivalence relation on $A$. 7. **Find all lines related to the line $y = 3x + 5$:** Two lines are parallel if they have the same slope. The slope of the line $y = 3x + 5$ is $3$. All lines parallel to it are of the form $$y = 3x + c$$ where $c$ is any real number. **Final answer:** The relation $R$ is an equivalence relation on the set of all lines $A$. The set of all lines related to $y = 3x + 5$ is $$\{y = 3x + c : c \in \mathbb{R}\}.$$