Simplifying Algebraic Expressions

Once variables enter the picture, expressions can get cluttered fast. You'll see something like $3x + 5 - 2x + 7$ and wonder where to even start. The good news: simplifying is mostly about recognizing patterns and applying a few basic rules. It's like organizing a messy desk — grouping similar items together and pitching the rest.

Simplifying isn't the same as solving. You're not finding a value for the variable; you're just rewriting the expression in its cleanest form. That makes it much easier to work with later, especially when you do need to solve an equation.

What Are Like Terms?

Like terms are terms that have the exact same variable part. The coefficients (the numbers in front) can be different, but the variable part must match exactly.

For example, $3x$ and $5x$ are like terms because they both have just $x$. You can combine them: $3x + 5x = 8x$. Think of it this way: if you have 3 apples and someone gives you 5 more apples, you now have 8 apples. Same logic.

But $3x$ and $3x^2$ are NOT like terms. One has $x$ and the other has $x^2$, which are completely different. You can't add 3 apples to 3 apple pies and get 6 of something — those are different things.

Here are more examples of like terms:

$7y$ and $-2y$ are like terms (both have $y$)
$4ab$ and $9ab$ are like terms (both have $ab$)
$x^2$ and $-5x^2$ are like terms (both have $x^2$)

And here are terms that are NOT alike:

$5x$ and $5y$ (different variables)
$3a$ and $3ab$ (different variable parts)
$2x^2$ and $2x^3$ (different exponents)

The constant terms (numbers without variables) are also like terms with each other. So $5$, $-3$, and $12$ can all be combined together.

Combining Like Terms

Once you've identified like terms, combining them is straightforward: add or subtract their coefficients and keep the variable part unchanged.

Example: Simplify $4x + 7x$.

Both terms have $x$, so they're like terms. Add the coefficients: $4 + 7 = 11$.

Answer: $11x$

Example: Simplify $9y - 5y$.

Both terms have $y$. Subtract the coefficients: $9 - 5 = 4$.

Answer: $4y$

Try one with multiple types of terms.

Example: Simplify $5x + 3y - 2x + 8y$.

First, identify the like terms. There are two $x$ terms ($5x$ and $-2x$) and two $y$ terms ($3y$ and $8y$).

Combine the $x$ terms: $5x - 2x = 3x$

Combine the $y$ terms: $3y + 8y = 11y$

Answer: $3x + 11y$

Notice that $3x$ and $11y$ can't be combined further because they have different variables. That's as simple as the expression gets.

Example: Simplify $6a + 4 - 2a + 9$.

The like terms here are $6a$ and $-2a$, plus the constants $4$ and $9$.

Combine the $a$ terms: $6a - 2a = 4a$

Combine the constants: $4 + 9 = 13$

Answer: $4a + 13$

Using the Distributive Property First

Sometimes you need to use the distributive property before you can combine like terms. The distributive property says $a(b + c) = ab + ac$.

Example: Simplify $3(x + 4) + 2x$.

First, distribute the 3: $$3(x + 4) = 3x + 12$$

The expression becomes $3x + 12 + 2x$.

Combine like terms: $3x + 2x = 5x$

Answer: $5x + 12$

Example: Simplify $5(2y - 3) - 4(y + 1)$.

Distribute the 5: $5(2y - 3) = 10y - 15$

Distribute the $-4$: $-4(y + 1) = -4y - 4$

That gives $10y - 15 - 4y - 4$.

Combine the $y$ terms: $10y - 4y = 6y$

Combine the constants: $-15 - 4 = -19$

Answer: $6y - 19$

Pay special attention when distributing negative numbers. Each term inside the parentheses gets multiplied by that negative, which can flip signs in ways that trip people up.

A More Complex Example

Here's a longer expression that uses both moves at once.

Example: Simplify $7x + 3(2x - 5) - 4x + 8$.

Step 1: Distribute the 3. $$3(2x - 5) = 6x - 15$$

Step 2: Rewrite the expression. $$7x + 6x - 15 - 4x + 8$$

Step 3: Group like terms (this is mental organization; you don't have to write it down). What do you think the final answer will be? Show answer$9x - 7$. Combining $x$ terms: $7x + 6x - 4x = 9x$. Combining constants: $-15 + 8 = -7$.

Why Simplifying Matters

It's fair to ask why simplifying matters if no variable is being solved for. The answer: simpler expressions are easier to work with in every way. They're faster to evaluate when you substitute a number for the variable, easier to solve when they're part of an equation, easier to graph, and easier to read.

Many problems also require simplification as a first step. Solving $3(x + 2) + 4x = 26$ starts with cleaning up the left side — you can't isolate $x$ until you do.

Try These

Simplify $8x + 5x$ Show answer$13x$
Simplify $12y - 7y + 3$ Show answer$5y + 3$
Simplify $4a + 6b - a + 2b$ Show answer$3a + 8b$
Simplify $2(x + 3) + 5x$ Show answerFirst distribute: $2x + 6 + 5x$, then combine: $7x + 6$
Simplify $6(m - 2) - 3(m + 1)$ Show answerDistribute: $6m - 12 - 3m - 3$, then combine: $3m - 15$
Simplify $5x + 3 - 2x + 7 + x$ Show answerCombine $x$ terms: $5x - 2x + x = 4x$. Combine constants: $3 + 7 = 10$. Answer: $4x + 10$

What's Next?

A few sign-related pitfalls trip students up most often. Unlike terms can't merge into a single term, so $3x + 2y$ stays as $3x + 2y$ — never $5xy$. When you distribute a negative, it touches every term inside the parentheses: $-2(x - 3) = -2x + 6$, not $-2x - 6$. And subtraction in the coefficient counts: $5x - 8x = -3x$, not $3x$.

Simplifying is the warm-up move for solving. The next stop is solving one-step equations, where you'll start undoing operations to isolate the variable on its own side of the equals sign.