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4.6: Simplify Variable Expressions Involving Integer Addition

Difficulty Level: At Grade Created by: CK-12
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Ashley has just started a new business making and selling granola bars. So far she has been selling the granola bars at soccer games on Saturdays. Ashley sold 10 bars in her first week of business and 25 bars in her second week of business. If \begin{align*}b\end{align*} represents the profit she makes from selling one bar, how could Ashley write and simplify a variable expression that represents her current profit after two weeks in business?

In this concept, you will learn how to simplify variable expressions involving integer addition.

Simplifying Variable Expressions Involving Integer Addition

An expression is a number phrase that contains numbers and operations.

Here are some examples of expressions:

  • \begin{align*}2x-4\end{align*}
  • \begin{align*}-15+7-1\end{align*}
  • \begin{align*}3x+5y-2\end{align*}
  • \begin{align*}5x+2x\end{align*}

A variable is a symbol or letter (often \begin{align*}x\end{align*} or \begin{align*}y\end{align*}) that is used to represent one or more numbers. A variable expression is an expression that includes variables.

Here are some examples of variable expressions:

  • \begin{align*}2x-4\end{align*}
  • \begin{align*}3x+5y-2\end{align*}
  • \begin{align*}5x+2x\end{align*}

In a variable expression, like terms are two terms that include the same variable. If a variable expression has like terms, it can be simplified by combining the like terms into one single term.

Here is an example.

Simplify \begin{align*}5x+2x\end{align*}.

In this variable expression, \begin{align*}5x\end{align*} and \begin{align*}2x\end{align*} are like terms because they include the same variable, \begin{align*}x\end{align*}.

To simplify the expression, focus on the coefficients, which are the numbers in front of the \begin{align*}x\end{align*}'s. In this case, your coefficients are 5 and 2. Add them up in order to simplify the variable expression.

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

Think of it as if you had 5 \begin{align*}x\end{align*}'s and 2 more \begin{align*}x\end{align*}'s, you'd have 7 \begin{align*}x\end{align*}'s total.

Here is another example.

\begin{align*}9y + 8x\end{align*}

This variable expression cannot be combined or simplified. It does not have like terms. The \begin{align*}x\end{align*} and the \begin{align*}y\end{align*} are different, so you cannot do anything with this expression. It is in simplest form.

You can use what you know about adding integers to help you to simplify variable expressions.

Let's look at one more example.

Find the sum of \begin{align*}-3t + 9t\end{align*}.

Since \begin{align*}-3t\end{align*} and \begin{align*}9t\end{align*} both have the same variable, they are like terms. Use what you know about how to add integers to help you add the terms.

Notice that the two like terms have different signs. So, your first step is to find the absolute values of both integers.  Then subtract the term whose integer has the lesser absolute value from the other term.

\begin{align*}|−3| = 3\end{align*} and \begin{align*}|9| = 9\end{align*}

\begin{align*}9 - 3 = 6\end{align*}

Your next step is to decide what the sign should be on your final answer. Your answer should take the sign of the original term that had the greater absolute value. Since 9 is greater than 3, and \begin{align*}9t\end{align*} has a positive sign, give the answer a positive sign.

The answer is \begin{align*}-3t + 9t = 6t\end{align*}.

Examples

Example 1

Earlier, you were given a problem about Ashley and her granola bar business.

Her first week in business she sold 10 bars and her second week in business she sold 25 bars. She wants to come up with a variable expression to represent her total profit so far.

If \begin{align*}b\end{align*} represents her profit from one bar, then in the first week her profit was \begin{align*}10b\end{align*} and in the second week her profit was \begin{align*}25b\end{align*}. To find her total profit, she wants to simplify 

\begin{align*}10b+25b\end{align*}

To simplify this expression, start by looking at the signs of the terms. \begin{align*}10b\end{align*} is positive and \begin{align*}25b\end{align*} is also positive. Because they have the same sign, you will need to add their absolute values.

\begin{align*}|10| = 10\end{align*} and \begin{align*}|25| = 25\end{align*}.

\begin{align*}10+25=35\end{align*}

Your next step is to decide what the sign should be on your final answer. Your answer should take the sign of the original terms. Since the original terms were positive, your final answer should be positive.

The answer is \begin{align*}10b+25b=35b\end{align*}.

Ashley's profit so far is \begin{align*}35b\end{align*}, where \begin{align*}b\end{align*} represents the profit she makes from selling one granola bar.

Example 2

Simplify \begin{align*}7z+(-3z)\end{align*}.

First, notice that both terms have the same variable, \begin{align*}z\end{align*}, so they are like terms.

Next, look at the signs of the terms. \begin{align*}7z\end{align*} is positive, but \begin{align*}-3z\end{align*} is negative. Because they have different signs, you will need to subtract their absolute values.

\begin{align*}|7| = 7\end{align*} and \begin{align*}|-3| = 3\end{align*}.

\begin{align*}7-3=4\end{align*}

Your next step is to decide what the sign should be on your final answer. Your answer should take the sign of the original term that had the greater absolute value. Since 7 is greater than 3, and \begin{align*}7z\end{align*} has a positive sign, give the answer a positive sign.

The answer is \begin{align*}7z+(-3z)=4z\end{align*}.

Example 3

Simplify \begin{align*}-8x+(-5x)\end{align*}.

First, notice that both terms have the same variable, \begin{align*}x\end{align*}, so they are like terms.

Next, look at the signs of the terms. \begin{align*}-8x\end{align*} is negative and \begin{align*}-5x\end{align*} is also  negative. Because they have the same sign, you will need to add their absolute values.

\begin{align*}|-8| = 8\end{align*} and \begin{align*}|-5| = 5\end{align*}.

 \begin{align*}8+5=13\end{align*}

Your next step is to decide what the sign should be on your final answer. Your answer should take the sign of the original terms. Since the original terms were negative, your final answer should be negative.

The answer is \begin{align*}-8x+(-5x)=-13x\end{align*}.

Example 4

Simplify \begin{align*}-19y+5y\end{align*}.

First, notice that both terms have the same variable, \begin{align*}y\end{align*}, so they are like terms.

Next, look at the signs of the terms. \begin{align*}-19y\end{align*} is negative, but \begin{align*}5y\end{align*} is positive. Because they have different signs, you will need to subtract their absolute values.

\begin{align*}|-19| = 19\end{align*} and \begin{align*}|5| = 5\end{align*}.

\begin{align*}19-5=14\end{align*}

Your next step is to decide what the sign should be on your final answer. Your answer should take the sign of the original term that had the greater absolute value. Since 19 is greater than 5, and \begin{align*}-19y\end{align*} has a negative sign, give the answer a negative sign.

The answer is \begin{align*}-19y+5y=-14y\end{align*}.

Example 5

Simplify \begin{align*}-6y+2y+(-3y)\end{align*}.

First, notice that all three terms have the same variable, \begin{align*}y\end{align*}, so they are all like terms.

Now, you will do this problem in two steps. Start by adding \begin{align*}-6y+2y\end{align*}.

Look at the signs of the terms. \begin{align*}-6y\end{align*} is negative, but \begin{align*}2y\end{align*} is positive. Because they have different signs, you will need to subtract their absolute values.

\begin{align*}|-6| = 6\end{align*} and \begin{align*}|2| = 2\end{align*}.

\begin{align*}6-2=4\end{align*}

Your next step is to decide what the sign should be on your result. Your result should take the sign of the original term that had the greater absolute value. Since 6 is greater than 2, and \begin{align*}-6y\end{align*} has a negative sign, give the result a negative sign.

\begin{align*}-6y+2y=-4y\end{align*}

Next, take \begin{align*}-4y\end{align*} and add to it the final term from the original expression, \begin{align*}-3y\end{align*}.

Again, look at the signs of the terms. \begin{align*}-4y\end{align*} is negative and \begin{align*}-3y\end{align*} is also negative. Because they have the same sign, you will need to add their absolute values.

\begin{align*}|-4| = 4\end{align*} and \begin{align*}|-3| = 3\end{align*}.

\begin{align*}4+3=7\end{align*}

Your next step is to decide what the sign should be on your final answer. Your answer should take the sign of the original terms. Since the original terms (\begin{align*}-4y\end{align*} and \begin{align*}-3y\end{align*}) were negative, your final answer should be negative.

\begin{align*}-4y+(-3y)=-7y\end{align*}

The answer is \begin{align*}-6y+2y+(-3y)=-7y\end{align*}.

Review

Simplify each variable expression.

  1. \begin{align*}7z+(-3z)\end{align*} 
  2. \begin{align*}17z+(-15z)\end{align*} 
  3. \begin{align*}5x+(-3x)\end{align*} 
  4. \begin{align*}8y+(2y)\end{align*} 
  5. \begin{align*}12x+(-13x)\end{align*} 
  6. \begin{align*}9z+(-9z)\end{align*} 
  7. \begin{align*}14a+(-3a)\end{align*} 
  8. \begin{align*}22y+(-33y)\end{align*} 
  9. \begin{align*}(-10d)+(-d)+2\end{align*} 
  10. \begin{align*}8x+(-4x)-5\end{align*} 
  11. \begin{align*}7y+(-3y)\end{align*} 
  12. \begin{align*}16x+(-22x)\end{align*} 
  13. \begin{align*}5a+(-a)+7a\end{align*} 

Solve each real-world problem.

  1. A plane is flying at an altitude that is 2, 500 feet above sea level. If the plane increases its altitude by 500 feet more, what will be its new altitude?
  2. The temperature on a mountaintop at midnight was \begin{align*}−8^\circ F\end{align*}. By 3:00 A.M., the temperature had risen by \begin{align*}3^\circ F\end{align*}. What is the temperature at 3:00 A.M.?

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.6.

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