Corey has a bowl of fruit that consists of 5 apples, 4 oranges, and 3 limes. Katelyn went to the farmer's market and picked up 2 apples, 5 limes, and an orange. How many apples, oranges, and limes do Corey and Katelyn have combined?

Combining like terms is much like grouping together different fruits, like apples and oranges.

### Combining Like Terms

Sometimes variables and numbers can be repeated within an expression. If the same variable is in an expression more than once, the occurrences can be combined by addition or subtraction. This process is called **combining like terms.**

Let's simplify each of the following expressions.

- Simplify \begin{align*}5x-12-3x+4\end{align*}.

Reorganize the expression to group together the \begin{align*}x\end{align*}’s and the numbers. You can either place the like terms next to each together or place parenthesis around the like terms.

\begin{align*}& 5x-12-3x+4\\ & 5x-3x-12+4 \ or \ (5x-3x)+(-12+4)\\ & 2x-8\end{align*}

Notice that the Greatest Common Factor (GCF) for \begin{align*}2x\end{align*} and 8 is 2. Therefore, you can use the Distributive Property to pull out the GCF to get \begin{align*}2(x-4)\end{align*}.

- Simplify \begin{align*}6a-5b+2a-10b+7\end{align*}.

Here there are two different variables, \begin{align*}a\end{align*} and \begin{align*}b\end{align*}. Even though they are both variables, they are *different* variables and cannot be combined. Group together the like terms.

\begin{align*}& 6a-5b+2a-10b+7\\ & (6a+2a)+(-5b-10b)+7\\ & (8a-15b+7)\end{align*}

There is only one number term, called the **constant**, so we leave it at the end. Also, in general, list the variables in alphabetical order.

- Simplify \begin{align*}w^2+9-4w^2+3w^4-7w-11\end{align*}.

Here we have one variable, but there are different powers (exponents). Like terms must have the same exponent in order to combine them.

\begin{align*}& w^2+9-4w^2+3w^4-7w-11\\ & 3w^4+(w^2-4w^2)-7w+(9-11)\\ & 3w^4-3w^2-7w-2\end{align*}

When writing an expression with different powers, list the powers from greatest to least, like above.

### Examples

#### Example 1

Earlier, you were asked to find the total number of apples, oranges, and limes Corey and Katelyn have all together.

Let's rewrite Corey's bowl of fruit as \begin{align*}5a+4o+3l\end{align*}, where \begin{align*}a\end{align*} represents apples, \begin{align*}o\end{align*} represents oranges, and \begin{align*}l\end{align*} represents limes. Then Katelyn's bowl of fruit can be represented using the same format as \begin{align*}2a+5l+o\end{align*}. Combining like terms, we have:

\begin{align*}(&5a+4o+3l)+(2a+5l+o)\\ (&5a+2a)+(4o+o)+(3l+5l)\\ &7a+5o+8l\end{align*}

Together they have 7 apples, 5 oranges, and 8 limes.

#### Example 2

Simplify the following expression: \begin{align*}6s-7t+12t-10s\end{align*}.

Combine the \begin{align*}s\end{align*}’s and the \begin{align*}t\end{align*}’s.

\begin{align*}& 6s-7t+12t-10s\\ & (6s-10s)+(-7t+12t)\\ & \text{-}4s+5t\end{align*}

Notice in that we did not write \begin{align*}(6s-10s)-(7t+12t)\end{align*} in the second step. This would lead us to an incorrect answer. Whenever grouping together like terms, if one is negative (or being subtracted), always change the operator to addition and make the subtracted number negative. In other words, when applying the commutative property to reorganize an expression, keep the sign to the left of a term with the term, then you can remove the extraneous +/- signs after you have everything sorted, like this:

\begin{align*}& 6s-7t+12t-10s\\
& (6s)+(-7t)+(+12t)+(-10s)\\
& (6s)+(-10s)+(+12t)+(-7t)\\
& 6s-10s+12t-7t\\
& \text{-}4s+5t\end{align*}

#### Example 3

Simplify the following expression: \begin{align*}7y^2-9x^2+y^2-14x+3x^2-4\end{align*}.

Group together the like terms.

\begin{align*}& 7y^2-9x^2+y^2-14x+3x^2-4\\ & (\text{-}9x^2+3x^2)+(7y^2+y^2)-14x-4\\ & \text{-}6x^2+8y^2-14x-4\end{align*}

### Review

Simplify the following expressions as much as possible. If the expression cannot be simplified, write “cannot be simplified.”

- \begin{align*}5b-15b+8d+7d\end{align*}
- \begin{align*}6-11c+5c-18\end{align*}
- \begin{align*}3g^2-7g^2+9+12\end{align*}
- \begin{align*}8u^2+5u-3u^2-9u+14\end{align*}
- \begin{align*}2a-5f\end{align*}
- \begin{align*}7p-p^2+9p+q^2-16-5q^2+6\end{align*}
- \begin{align*}20x-6-13x+19\end{align*}
- \begin{align*}8n-2-5n^2+9n+14\end{align*}

Find the GCF of the following expressions and use the Distributive Property to simplify each one.

- \begin{align*}6a-18\end{align*}
- \begin{align*}9x^2-15\end{align*}
- \begin{align*}14d+7\end{align*}
- \begin{align*}3x-24y+21\end{align*}

**Challenge** We can also use the Distributive Property and GCF to pull out common variables from an expression. Find the GCF and use the Distributive Property to simplify the following expressions.

- \begin{align*}2b^2-5b\end{align*}
- \begin{align*}m^3-6m^2+11m\end{align*}
- \begin{align*}4y^4-12y^3-8y^2\end{align*}

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 1.5