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# 4.6: Simplify Variable Expressions Involving Integer Addition

Difficulty Level: At Grade Created by: CK-12

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 b\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.

### Guidance

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

Here are some examples of expressions:

• 2x4\begin{align*}2x-4\end{align*}
• 15+71\begin{align*}-15+7-1\end{align*}
• 3x+5y2\begin{align*}3x+5y-2\end{align*}
• 5x+2x\begin{align*}5x+2x\end{align*}

A variable is a symbol or letter (often x\begin{align*}x\end{align*} or y\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:

• 2x4\begin{align*}2x-4\end{align*}
• 3x+5y2\begin{align*}3x+5y-2\end{align*}
• 5x+2x\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 5x+2x\begin{align*}5x+2x\end{align*}.

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

To simplify the expression, focus on the coefficients, which are the numbers in front of the x\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.

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

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

Here is another example.

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

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

Let's look at one more example.

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

Since 3t\begin{align*}-3t\end{align*} and 9t\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.

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

93=6\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 9t\begin{align*}9t\end{align*} has a positive sign, give the answer a positive sign.

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

### Guided Practice

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

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

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

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

73=4\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 7z\begin{align*}7z\end{align*} has a positive sign, give the answer a positive sign.

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

### Examples

#### Example 1

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

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

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

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

8+5=13\begin{align*}8+5=13\end{align*}

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

#### Example 2

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

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

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

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

195=14\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 19y\begin{align*}-19y\end{align*} has a negative sign, give the answer a negative sign.

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

#### Example 3

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

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

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

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

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

62=4\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 6y\begin{align*}-6y\end{align*} has a negative sign, give the result a negative sign.

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

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

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

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

4+3=7\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 (4y\begin{align*}-4y\end{align*} and 3y\begin{align*}-3y\end{align*}) were negative, your final answer should be negative.

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

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

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

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

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

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

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

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

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

### Explore More

Simplify each variable expression.

1. 7z+(3z)\begin{align*}7z + (−3z)\end{align*}

2. 17z+(15z)\begin{align*}17z + (−15z)\end{align*}

3. 5x+(3x)\begin{align*}5x + (−3x)\end{align*}

4. 8y+(2y)\begin{align*}8y + (2y)\end{align*}

5. 12x+(13x)\begin{align*}12x + (-13x)\end{align*}

6. 9z+(9z)\begin{align*}9z + (−9z)\end{align*}

7. 14a+(3a)\begin{align*}14a + (-3a)\end{align*}

8. 22y+(33y)\begin{align*}22y + (-33y)\end{align*}

9. (10d)+(d)+2\begin{align*}(-10d) + (-d) + 2\end{align*}

10. 8x+(4x)5\begin{align*}8x + (-4x) - 5\end{align*}

11. 7y+(3y)\begin{align*}7y + (-3y)\end{align*}

12. 16x+(22x)\begin{align*}16x + (-22x)\end{align*}

13. 5a+(a)+7a\begin{align*}5a + (-a) + 7a\end{align*}

Solve each real-world problem.

14. 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?

15. The temperature on a mountaintop at midnight was 8F\begin{align*}−8^\circ F\end{align*}. By 3:00 A.M., the temperature had risen by 3F\begin{align*}3^\circ F\end{align*}. What is the temperature at 3:00 A.M.?

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