Identifying Equivalent Expressions

Have you ever noticed that two math expressions can look completely different but actually mean the exact same thing? These are called equivalent expressions. No matter what value you plug in for the variable, equivalent expressions will always give you the same result.

There are two main ways to check if expressions are equivalent: simplifying them using math properties, or substituting a value for the variable.

Method 1: Simplifying Using Properties

You can use the distributive, commutative, and associative properties to rewrite an expression. If you can rewrite one expression to look exactly like the other, they are equivalent.

Example: Are $3(x + 2)$ and $3x + 6$ equivalent?

Using the distributive property, we multiply the $3$ by everything inside the parentheses: $3(x + 2) = 3(x) + 3(2)$ $3(x) + 3(2) = 3x + 6$

Since simplifying $3(x + 2)$ gives us exactly $3x + 6$, the two expressions are equivalent!

We can also go the other way by factoring.

Example: Rewrite $6x + 9$. Both terms share a common factor of $3$. $6x + 9 = 3(2x) + 3(3) = 3(2x + 3)$

Method 2: Checking by Substitution

If you aren't sure how to simplify, you can pick a random number for your variable, plug it into both expressions, and see if the answers match. It's usually best to pick an easy number like $2$ or $3$ (avoid $0$ or $1$ as they can sometimes lead to false matches).

Example: Are $2(x - 1) + 3$ and $2x + 1$ equivalent?

Let's substitute $x = 4$ into both expressions.

First expression: $2(4 - 1) + 3$ $= 2(3) + 3$ $= 6 + 3 = 9$

Second expression: $2(4) + 1$ $= 8 + 1 = 9$

Since both expressions equal $9$ when $x = 4$, they are equivalent! You can also double-check this by simplifying the first expression: $2x - 2 + 3$, which combines to $2x + 1$.