Combining Like Terms

Algebraic expressions often contain several terms that can be merged into fewer, simpler terms. The process of merging those terms is called combining like terms, and it is one of the most frequently used simplification skills in all of algebra. If you can add $3 + 5$ to get $8$, you can combine $3x + 5x$ to get $8x$ — the logic is the same.

What Are Like Terms?

Like terms are terms that have exactly the same variable(s) raised to exactly the same exponent(s). Only the coefficients may differ.

The key rule: same variable(s), same exponent(s). The coefficient does not matter when deciding whether terms are alike — it only matters when you combine them.

How to Combine Like Terms

To combine like terms, add or subtract their coefficients and keep the variable part unchanged.

Example 1: Simplify $7x + 3x$

Both terms have the variable $x$ to the first power, so they are like terms.

$7x + 3x = (7 + 3)x = 10x$

Think of it this way: 7 apples plus 3 apples equals 10 apples. The “apples” ($x$) do not change.

Example 2: Simplify $9y - 4y$

$9y - 4y = (9 - 4)y = 5y$

Example 3: Simplify $5a + 3b + 2a - b$

Group like terms together:

$5a + 2a + 3b - b$

Combine the $a$ terms and the $b$ terms separately:

$(5 + 2)a + (3 - 1)b = 7a + 2b$

Answer: $7a + 2b$

Example 4: Simplify $3x^2 + 5x - 2x^2 + x - 4$

Identify like terms:

• $x^2$ terms: $3x^2$ and $-2x^2$
• $x$ terms: $5x$ and $x$ (remember, $x = 1x$)
• Constants: $-4$

Combine each group:

$(3 - 2)x^2 + (5 + 1)x - 4 = x^2 + 6x - 4$

Answer: $x^2 + 6x - 4$

Rearranging Before Combining

It often helps to rearrange an expression so that like terms sit next to each other. This reduces errors.

Example 5: Simplify $2m + 7 - 5m + 3$

Rearrange to group like terms:

$2m - 5m + 7 + 3$

Combine:

$-3m + 10$

Answer: $-3m + 10$

Example 6: Simplify $4x + 2y - x + 3y - 6$

Rearrange:

$4x - x + 2y + 3y - 6$

Combine:

$3x + 5y - 6$

Answer: $3x + 5y - 6$

Expressions with Multiple Variables

When an expression has several different variables, combine only the terms that share the same variable.

Example 7: Simplify $6a - 2b + 3c + 4b - a + c$

Group by variable:

$6a - a - 2b + 4b + 3c + c$

Combine:

$5a + 2b + 4c$

Answer: $5a + 2b + 4c$

Real-World Application: Carpentry — Material Estimates

A carpenter is building two identical bookshelves and one desk. Each bookshelf requires $4x$ feet of lumber for the shelves and $2x$ feet for the frame. The desk requires $3x$ feet for the top and $x$ feet for the legs, where $x$ depends on the size grade selected.

Total lumber needed:

$\text{Bookshelves: } 2(4x + 2x) = 2(6x) = 12x$

$\text{Desk: } 3x + x = 4x$

$\text{Total: } 12x + 4x = 16x$

If the customer picks the medium size grade where $x = 3$ feet:

$16(3) = 48 \text{ feet of lumber}$

Combining like terms first made the final calculation much simpler — one multiplication instead of many separate ones.

Real-World Application: Retail — Inventory Expression

A retail manager tracks weekly stock changes. On Monday the store receives $n$ boxes of product A and $2n$ boxes of product B. On Wednesday, they return $3$ boxes of A and receive $n$ more boxes of B. The net stock expression is:

$n + 2n - 3 + n$

Combine the $n$ terms:

$4n - 3$

This simplified expression tells the manager that total inventory change depends on four times the base shipment size, minus 3 boxes.

Common Mistakes to Avoid

1. Combining unlike terms. You cannot add $3x + 2y$ to get $5xy$. Different variables stay separate: $3x + 2y$ is already simplified.

2. Forgetting the invisible coefficient of 1. In $x + 5x$, the first term has a coefficient of $1$: $1x + 5x = 6x$

3. Mixing up exponents. $x$ and $x^2$ are not like terms. You cannot combine $4x + 3x^2$ — the expression stays as $3x^2 + 4x$.

4. Dropping the sign. When rearranging terms, the sign in front of each term travels with it. In $8 - 3x + 2$, the $-3x$ keeps its negative sign: $-3x + 8 + 2 = -3x + 10$.

5. Changing the variable when combining. The variable part does not change — only the coefficient changes. $5x + 3x = 8x$, not $8x^2$.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Like terms share the same variable(s) with the same exponent(s)
• To combine like terms, add or subtract the coefficients — the variable part stays unchanged
• Rearranging the expression to group like terms together reduces mistakes
• Unlike terms (different variables or different exponents) cannot be combined — they stay separate
• A lone variable like $x$ has a coefficient of $1$; a negative lone variable like $-x$ has a coefficient of $-1$
• In real-world applications, combining like terms simplifies formulas before you plug in numbers, saving time and reducing errors

Return to Pre-Algebra for more topics in this section.