20.1: Why is it True?
Explain why each statement is true.
- $5+2+3=5+(2+3)$
- $9a$ is equivalent to $11a-2a$.
- $7a+4-2a$ is equivalent to $7a+\text-2a+4$.
- $8a-(8a-8)$ is equivalent to 8.
Let's see how we can tell that expressions are equivalent.
Explain why each statement is true.
Diego and Jada are both trying to write an expression with fewer terms that is equivalent to $$7a + 5b - 3a + 4b$$
We can show expressions are equivalent by writing out all the variables. Explain why the expression on each row (after the first row) is equivalent to the expression on the row before it. $$7a+5b-3a+4b$$ $$(a+a+a+a+a+a+a) + (b+b+b+b+b) - (a+a+a) + (b+b+b+b)$$ $$(a+a+a+a) + (a+a+a) + (b+b+b+b+b) - (a+a+a) + (b+b+b+b)$$ $$(a+a+a+a) + (b+b+b+b+b) + (a+a+a) - (a+a+a) + (b+b+b+b)$$ $$(a+a+a+a) + (b+b+b+b+b) + (b+b+b+b)$$ $$(a+a+a+a) + (b+b+b+b+b+b+b+b+b)$$ $$4a + 9b$$
Here is another way we can rewrite the expressions. Explain why the expression on each row (after the first row) is equivalent to the expression on the row before it. $$7a+5b-3a+4b$$ $$7a+5b+(\text-3a)+4b$$ $$7a+(\text-3a)+5b+4b$$ $$(7+\text-3)a+(5+4)b$$ $$4a+9b$$
Follow the instructions for a number puzzle:
Replace each ? with an expression that will make the left side of the equation equivalent to the right side.
Set A
Check your results with your partner and resolve any disagreements. Then move on to Set B.
Set B
There are many ways to write equivalent expressions that may look very different from each other. We have several tools to find out if two expressions are equivalent.
\(\begin{align} 2(\text-3+x)+8\\ \text-6+2x+8 & \quad\text{by the distributive property}\\ 2x+\text-6+8 & \quad\text{by the commutative property}\\ 2x+(\text-6+8) & \quad\text{by the associative property} \\ 2x+2 \\ \end{align}\)