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7.4: Simplify Sums or Differences of Single Variable Expressions

Difficulty Level: At Grade Created by: CK-12
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Marc is heading to his local ice cream store. This particular store doesn't have a lot of flavor choices, but they have the BEST vanilla ice cream around. Marc has taken orders from several of his neighbors, too, and has written them all down so that he can keep track of who wants what.

The Johnsons - two vanilla ice cream cones

The Mumfords - three vanilla ice cream cones

Jill Stales - one vanilla ice cream cone

The Porters - three vanilla ice cream cones

How can Marc write this information as an algebraic expression and then simplify it?

In this concept, you will learn how to simplify single variable expressions.

Simplifying Sums or Differences of Single Variable Expressions

If an expression has only numbers, you can calculate its numerical value. However, if an expression includes variables, it is helpful to simplify the expression.

Look at this example.

Simplify the expression \begin{align*}6a + 3a\end{align*}.

When adding expressions with variables, it is important to remember that only like terms can be combined. For example, \begin{align*}6a\end{align*} and \begin{align*}3a\end{align*} are like terms because both terms include the variable \begin{align*}a\end{align*}. So, you can combine them.

\begin{align*}\begin{array}{rcl} && 6a+3a \\ && 9a \end{array}\end{align*}

Here is another example.

Simplify \begin{align*}6a + 3\end{align*}.

\begin{align*}6a\end{align*} and \begin{align*}3\end{align*} are not like terms because only one term includes the variable \begin{align*}a\end{align*}. So, you cannot combine them. The expression \begin{align*}6a + 3\end{align*} cannot be simplified any further.

Here is another example.

Simplify \begin{align*}15d -2d\end{align*}.

Since \begin{align*}15d\end{align*} and \begin{align*}2d\end{align*} both have the same variable, they are like terms. To find the difference, subtract the numerical parts of the terms the same way you would subtract any numbers.

\begin{align*}15d - 2d = 13d\end{align*}

The answer is \begin{align*}13d\end{align*}.

Here is an example using decimals.

Simplify \begin{align*}0.4x + 1.3x\end{align*}.

Since \begin{align*}0.4x\end{align*} and \begin{align*}1.3x\end{align*} both have the same variable, they are like terms. To find the sum, add the numerical parts of the terms the same way you would add any decimals.

\begin{align*}0.4x + 1.3x = 1.7x\end{align*}

The answer is \begin{align*}1.7x\end{align*}.


Example 1

Earlier, you were given a problem about Marc and all of his ice cream orders.

Marc needs to write an expression to simplify his order: two vanillas, three vanillas, one vanilla, and three vanillas.

Let \begin{align*}v\end{align*} represent an ice cream cone. Then you can represent this situation as a single variable expression.

\begin{align*}2v + 3v + v + 3v\end{align*}

Looking at this expression, you will see that the variables are all the same. Therefore, simply add the numerical part of each term.

\begin{align*}2v + 3v + v + 3v = 9v\end{align*}

The answer is \begin{align*}9v\end{align*}.

Example 2

Simplify the expression.

\begin{align*}5a + 4a- 2a + 6a\end{align*}

To simplify this expression, follow the order of operations and combine like terms in order from left to right. Here is what the expression looks like after the first two terms have been combined.

\begin{align*}9a- 2a + 6a\end{align*}

Next, perform the subtraction to get: \begin{align*}7a + 6a\end{align*}

Finally, add the terms.

\begin{align*}7a + 6a=13a\end{align*}

The answer is \begin{align*}13a\end{align*}.

Simplify each sum or difference when possible.

Example 3

\begin{align*}3a + 12a\end{align*}

These are like terms, so add the numerical parts together.

The answer is \begin{align*}15a\end{align*}.

Example 4

\begin{align*}16x- 2x\end{align*}

These are like terms, so subtract the numerical parts.

The answer is \begin{align*}14x\end{align*}.

Example 5

\begin{align*}7y + 2x\end{align*}

These are not like terms.

The terms are not alike so you cannot combine them. The expression is in the simplest form already.

The answer is \begin{align*}7y+2x\end{align*}.


Simplify each sum or difference by combining like terms.

  1. \begin{align*}6a + 7a\end{align*}
  2. \begin{align*}7x-2x\end{align*}
  3. \begin{align*}6y + 12y\end{align*}
  4. \begin{align*}8a + 12a\end{align*}
  5. \begin{align*}12y-7y\end{align*}
  6. \begin{align*}8a + 15a\end{align*}
  7. \begin{align*}13b- 9b\end{align*}
  8. \begin{align*}22x + 19x\end{align*}
  9. \begin{align*}45y-12y\end{align*}
  10. \begin{align*}16a + 18a + 9a\end{align*}
  11. \begin{align*}14x-6x + 2x\end{align*}
  12. \begin{align*}21a + 14a-15a\end{align*} 
  13. \begin{align*}33b + 13b + 8b\end{align*}
  14. \begin{align*}45x + 67x- 29x\end{align*}
  15. \begin{align*}92y + 6y-54y\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 7.4.


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Difference The result of a subtraction operation is called a difference.
Expression An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.
Simplify To simplify means to rewrite an expression to make it as "simple" as possible. You can simplify by removing parentheses, combining like terms, or reducing fractions.
Sum The sum is the result after two or more amounts have been added together.

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