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Sums and Differences of Single Variable Expressions

Simplify expressions by combining like terms.

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Sums and Differences of Single Variable Expressions
License: CC BY-NC 3.0

Jacob’s mother has put him in charge of planning a surprise party for his little brother, Kyle. Jacob plans to serve cake and ice-cream cups. He does not know how much each ice-cream cup will cost. He has invited 24 of Kyle’s friends and 6 other family members. Jacob needs to figure out how much the ice-cream cups will cost so he can get the money from his mother. How can he formulate a single variable expression to determine the total costs of the ice-cream cups?

In this concept, you will learn to simplify single-variable expressions that include sums and differences.

Simplifying Expressions Involving Sums and Differences

In mathematics, simplifying does not mean solving something. Simplifying means making something smaller.

Sometimes, you will be given an expression using variables where there is more than one term. A term is a number with a variable. Here is an example of a term.

\begin{align*}4x\end{align*}4x

You have not been given a value for \begin{align*}x\end{align*}, so this term cannot be simplified. If you had been given a value for \begin{align*}x\end{align*}, then you could evaluate the expression.

When there is more than one like term in an expression, you can simplify the expression. A like term means that the terms in question use the same variable.

\begin{align*}\underline{4x}\end{align*} and \begin{align*}\underline{5x}\end{align*} are like terms. They both have \begin{align*}\underline{x}\end{align*} as the variable. They are alike.

\begin{align*}\underline{6x}\end{align*} and \begin{align*}\underline{2y}\end{align*} are not like terms. One has an \begin{align*}\underline{x}\end{align*} and one has a \begin{align*}\underline{y}\end{align*}. They are not alike.

Expressions with like terms can be simplified.

You can also simplify the sums and differences of expressions with like terms. Let’s start with sums.

Here is an expression.

\begin{align*}5x + 7x\end{align*}

First, look to see if these terms are alike. Both of them have an \begin{align*}x\end{align*}, so they are alike.

Next, simplify them by adding the numerical part of the terms together. The \begin{align*}x\end{align*} stays the same.

\begin{align*}\begin{array}{rcl} && 5x + 7x \\ && 5 + 7 = 12 \\ &\text{So }, & 5x + 7x = 12x \end{array}\end{align*}

You can think of the \begin{align*}x\end{align*} as a label that lets you know that the terms are alike.

Here is another expression.

\begin{align*}7x + 2x + 5y\end{align*}

First, look to see if the terms are alike.

Two of the terms have \begin{align*}x\end{align*}‘s and one has a \begin{align*}y\end{align*}. The two with the \begin{align*}x\end{align*}‘s are alike. The one with the \begin{align*}y\end{align*} is not alike. You can simplify the ones with the \begin{align*}x\end{align*}‘s by adding the numerical part of the terms.

Next, simplify the like terms.

\begin{align*}7x + 2x = 9x\end{align*}

Then, because \begin{align*}5y\end{align*} can’t be simplified, it stays the same.

The answer is \begin{align*} 9x + 5y\end{align*}.

You can also simplify expressions with differences and like terms.

Here is an expression.

\begin{align*}9y- 2y\end{align*}

First, you can see that these terms are alike because they both have \begin{align*}y\end{align*}’s. Simplify the expression by subtracting the numerical part of the terms.

\begin{align*}\begin{array}{rcl} 9 - 2 &=& 7 \\ \text{So}, \ 9y - 2y &=& 7y \end{array}\end{align*}

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

Sometimes you can combine like terms that have both sums and differences in the same problem.

Here is another expression. 

\begin{align*}8x- 3x + 2y + 4y\end{align*}

First, group and simplify the like terms.

\begin{align*}\begin{array}{rcl} 8x- 3x = 5x \\ 2y + 4y = 6y \end{array}\end{align*}

Next, put it all together.

The answer is \begin{align*}5x + 6y\end{align*}.

Remember that you can only combine terms that are alike.

Examples

Example 1

Earlier, you were given a problem about Kyle’s surprise birthday party.

Jacob needs to know the cost of 24 ice-cream cups for Kyle’s friends and 6 ice-cream cups for the invited family members.

First, identify the numbers.

24 and 6

Next, make \begin{align*}x\end{align*} represent the cost of each ice-cream cup.

\begin{align*}x\end{align*}

Then, identify the key word “and” which means addition. 

\begin{align*}+\end{align*}

Now, write an expression representing the situation. \begin{align*}24x + 6x\end{align*}

Finally, simplify the expression by adding the numerical value of like terms.

\begin{align*}\begin{array}{rcl} 24 + 6 &=& 25 \\ \text{So}, 24x + 6x &=& 30x \end{array}\end{align*}

The answer is \begin{align*}30x\end{align*}.

Jacob will need to multiply the price of an ice-cream cup by 30 to determine the total cost.

Example 2

Simplify the following expression. 

\begin{align*}5x + 2x-1x + 6y- 4y\end{align*}

First, combine the like terms.

\begin{align*}& 5x + 2x- 1x\\ & 6y - 4y\end{align*} 

Next, simplify the like terms.

\begin{align*}\begin{array}{rcl} 5x + 2x- 1x &=& 6x \\ 6y-4y &=& 2y \end{array}\end{align*}

Then, put those simplified terms together.

\begin{align*}6x + 2y\end{align*}

The answer is \begin{align*}6x + 2y\end{align*}.

Example 3

Simplify the following expression.

\begin{align*}7z + 2z + 4z\end{align*}

First, look to see if these terms are alike. They all have an \begin{align*}z\end{align*}, so they are alike. 

Next, simplify them by adding the numerical part of the terms together. The \begin{align*}z\end{align*} stays the same.

\begin{align*}7 + 2 + 4 = 13\end{align*}

Then, replace the \begin{align*}z\end{align*}.

So, \begin{align*}7z + 2z + 4z = 13z\end{align*}

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

Example 4

Simplify the following expression.

\begin{align*}25y-13y\end{align*}

First, look to see if these terms are alike. Both have a \begin{align*}y\end{align*}, so they are alike.

Next, simplify them by adding the numerical part of the terms together. The \begin{align*}y\end{align*} stays the same.

\begin{align*}25 - 13 = 12\end{align*}

Then, replace the \begin{align*}y\end{align*}

\begin{align*}12y\end{align*}

The answer is \begin{align*}12y\end{align*}.

Example 5

Simplify the following expression.

\begin{align*}7x + 2x + 4a\end{align*}

First, combine the like terms.

\begin{align*}& 7x + 2x\\ & 4a\end{align*}

Next, simplify the like terms by adding the numerical coefficients.

\begin{align*}7 + 2 = 9\end{align*}

Then, replace the \begin{align*}x\end{align*}.

\begin{align*}& 9x\\ & 4a\end{align*}

The answer is \begin{align*}9x + 4a\end{align*}.

Review

Simplify the following expressions by combining like terms. If the expression is already in simplest form, write “already in simplest form.”

  1. \begin{align*}4x + 6x\end{align*}
  2. \begin{align*}8y + 5y\end{align*}
  3. \begin{align*}9z + 2z\end{align*}
  4. \begin{align*}8x + 2y\end{align*}
  5. \begin{align*}7y + 3y + 2x\end{align*}
  6. \begin{align*}9x-x\end{align*}
  7. \begin{align*}12y- 3y\end{align*} 
  8. \begin{align*}22x-2y\end{align*}
  9. \begin{align*}78x- 10x\end{align*}
  10. \begin{align*}22y-4y\end{align*}
  11. \begin{align*}16x- 5x + 1x- 12y + 2y\end{align*}
  12. \begin{align*}26x- 15x + 12x-14y + 2y\end{align*}
  13. \begin{align*}36x- 5x + 11x- 1x + 2y\end{align*}
  14. \begin{align*}26x- 25x + 12x- 13y + 2y\end{align*}
  15. \begin{align*}29x- 25x + 18x- 12x + 12y + 3y\end{align*} 

Review (Answers)

To see the Review answers, open this PDF file and look for section 12.4. 

Resources

 

Vocabulary

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.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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