Equivalent Expressions and Combining Like Terms

In algebra, an expression is a mathematical phrase that can contain numbers, variables, and operators. To make working with expressions easier, we often simplify them by combining like terms and creating equivalent expressions.

What Are Like Terms?

Like terms are terms in an expression that have the exact same variables raised to the exact same powers. The numbers in front of the variables (called coefficients) can be different.

• $3x$ and $5x$ are like terms because they both share the variable $x$.
• $2a$ and $7b$ are not like terms because they have different variables.
• Plain numbers, also known as constants (like $3$, $10$, or $-1$), are always like terms with each other.

How to Combine Like Terms

To combine like terms, you simply add or subtract their coefficients and keep the variable exactly the same. Think of it like counting objects: $3$ apples plus $5$ apples equals $8$ apples ($3x + 5x = 8x$).

Example 1: Simplify $4x + 3 + 2x - 1$

1. Identify and group the like terms together: $(4x + 2x) + (3 - 1)$.
2. Add or subtract the coefficients: $4x + 2x = 6x$.
3. Combine the constants: $3 - 1 = 2$.
4. The simplified expression is $6x + 2$.

Example 2: Combine like terms in $5a + 2b + 3a - b$

1. Group the like terms. Remember that subtracting $b$ is the same as subtracting $1b$.
2. Group the $a$ terms: $5a + 3a = 8a$.
3. Group the $b$ terms: $2b - 1b = 1b$ (or just $b$).
4. The simplified expression is $8a + b$.

Equivalent Expressions

Equivalent expressions are expressions that look different but have the exact same mathematical value for any number you substitute for the variable.

Combining like terms is one way to create equivalent expressions. Another common method is using the Distributive Property, which involves multiplying a number outside of parentheses by every term inside the parentheses.

Example 3: Are $3(x + 2)$ and $3x + 6$ equivalent? Let's use the distributive property to expand the first expression:

$3(x + 2) = (3 \times x) + (3 \times 2)$

$3(x + 2) = 3x + 6$

Because simplifying $3(x + 2)$ gives us $3x + 6$, yes, these two expressions are perfectly equivalent!