1. **Stating the problem:** We are given several algebraic expressions involving addition: $b + c = d$ $b + (c + d) = (b + c) + d$ $b + c = c + d$ $b + c = c + b$ We want to understand these expressions and the properties of addition they illustrate. 2. **Relevant properties of addition:** - **Associative property:** For any numbers $a, b, c$, we have $$a + (b + c) = (a + b) + c$$ - **Commutative property:** For any numbers $a, b$, we have $$a + b = b + a$$ 3. **Analyzing each expression:** - $b + c = d$: This defines $d$ as the sum of $b$ and $c$. - $b + (c + d) = (b + c) + d$: This is the associative property of addition. - $b + c = c + d$: This equation can be rewritten using the definition $d = b + c$ as $b + c = c + (b + c)$, which is generally not true unless $b=0$. - $b + c = c + b$: This is the commutative property of addition. 4. **Summary:** The key properties illustrated are the associative and commutative properties of addition. 5. **Final answer:** The expressions show that addition is associative and commutative, meaning the grouping and order of addition do not affect the sum.