We can use tape diagrams to see when expressions are equal. For example, the expressions $x+9$ and $4x$ are equal when $x$ is 3, but they are not equal for other values of $x$. 

8 tape diagrams on a grid with matching expressions. First diagram composed of 1 square unit labeled x and 9 square units combined which are blank, matched with x+9 when x=1. Second diagram composed of 4 square units each labeled x matched with 4x when x=1. Third diagram composed 2 combined square units labeled x and 9 combined square units blank, matched with x+9 when x=2. Fourth diagram composed of 2 combined square units labeled x created 4 total times, matched with 4x when x=2. Fifth diagram composed 3 combined square units labeled x and 9 combined square units blank, matched with x+9 when x=3. Sixth diagram composed 3 combined square units labeled x created 4 total times, matched with 4x when x=3. Seventh diagram composed of 4 combined square units labeled x and 9 combined square units blank, matched with x+9 when x=4. Eighth diagram composed of 4 combined square units labeled x created 4 total times, matched with 4x when x=4.

Sometimes two expressions are equal for only one particular value of their variable. Other times, they seem to be equal no matter what the value of the variable.

Expressions that are always equal for the same value of their variable are called equivalent expressions. However, it would be impossible to test every possible value of the variable. How can we know for sure that expressions are equivalent?

We can use the meaning of operations and properties of operations to know that expressions are equivalent. Here are some examples:

• $x+3$ is equivalent to $3+x$ because of the commutative property of addition. The order of the values being added doesn’t affect the sum.
• $4\boldcdot {y}$ is equivalent to $y\boldcdot 4$ because of the commutative property of multiplication. The order of the factors doesn’t affect the product.
• $a+a+a+a+a$ is equivalent to $5\boldcdot {a}$ because adding 5 copies of something is the same as multiplying it by 5.
• $b\div3$ is equivalent to $b \boldcdot {\frac13}$ because dividing by a number is the same as multiplying by its reciprocal.

In the coming lessons, we will see how another property, the distributive property, can show that expressions are equivalent.