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Simplify Variable Expressions Involving Multiple Operations

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Have you ever calculated train fare? Take a look at this dilemma.

Kara is very excited that she has figured out an expression to calculate the total train fare. She heads downstairs and shares her findings with her Grandpa.

.85x

“See Grandpa, if I put the number of rides in for x , then we can figure out the total cost,” Kara explains.

“That is great work Kara, but what about me? That will work great when it is just you and Marc, but I am riding with you today. Seniors ride for .60 per ride.”

Wow! Kara hadn’t even thought of that. Now she has a whole new dilemma.

“I’ve got it,” she said writing some notes on a piece of paper.

Kara began to write an expression and then she thought that she could combine the terms of the expression.

Do you have an idea what Kara is writing? Think about the expression that she wrote for teen fares and train rides. Now we are going to add to that and simplify. This Concept will teach you all about how this works. Focus on the information in the Concept and you will see this problem again at the end of it.

Guidance

Sometimes, you may need to simplify algebraic expressions that involve more than one operation. Use what you know about simplifying sums, differences, products, or quotients of algebraic expressions to help you do this.

When evaluating expressions, it is also important to keep in mind the order of operations . Let's review this order below.

  • First, do the computation inside grouping symbols, such as parentheses.
  • Second, evaluate any exponents.
  • Third, multiply and divide in order from left to right.
  • Finally, add and subtract in order from left to right.

That’s good because the order of operations is always useful in mathematics!! Now let’s look at an example.

Now let's apply this information.

Simplify this expression 7n+8n \cdot 3

According to the order of operations, you should multiply before you add.

7n+8n \cdot 3=7n+(8n \cdot 3) .

Separate out the factors and use the commutative property to help you multiply.

7n+(8n \cdot 3)=7n+(8 \cdot n \cdot 3)+7n+(8 \cdot 3 \cdot n)=7n+(24 \cdot n)=7n+24n

Since 7n and 24n are like terms, add them.

7n+24n=31n .

The answer is 31n .

Simplify this expression 10 p-7p+8p \div 2p .

According to the order of operations, you should divide before you subtract or add.

10p-7p+8p \div 2p=10p-7p+(8p \div 2p) .

It may help you to rewrite the division as \frac{8p}{2p} and then separate out the numbers and variables like this.

10p-7p+ \left(\frac{8p}{2p}\right)=10p-7p+\left(\frac{8 \cdot p}{2 \cdot p}\right)=10p-7p+\left(\frac{8}{2} \cdot \frac{p}{p}\right)=10p-7p+(4 \cdot 1)=10p-7p+4

The order of operations says to add and subtract in order from left to right. So, subtract the like terms 10p and 7p next.

10p-7p+4=3p+4 .

Since 3p and 4 are not like terms, those terms cannot be combined. So, the expression cannot be simplified further.

The expression, when simplified, is 3p+4 . This is our final answer.

Simplify each expression.

Example A

4a+9a - 7

Solution: 13a - 7

Example B

\frac{14x}{2}+9x

Solution: 16x

Example C

6b-2b+5b-8

Solution: 9b-8

Here is the original problem once again.

Kara is very excited that she has figured out an expression to figure out the total train fare. She heads downstairs and shares her findings with her Grandpa.

.85x

“See Grandpa, if I put the number of rides in for x , then we can figure out the total cost,” Kara explains.

“That is great work Kara, but what about me? That will work great when it is just you and Marc, but I am riding with you today. Seniors ride for .60 per ride.”

Wow! Kara hadn’t even thought of that. Now she has a whole new dilemma.

“I’ve got it,” she said writing some notes on a piece of paper.

Kara began to write an expression and then she thought that she could combine the terms of the expression.

To write an expression that includes Grandpa, Kara can begin with the first expression that she wrote.

.85x this accounts for the teen fare and the number of rides which is unknown so we use x .

Next, we have to include Grandpa. Seniors ride for .60 per ride. The number of rides is still unknown, so we can use x for that too.

.60x

Since they will be riding together, we can add the two terms.

.85x+.60x

Next, we can simplify the expression.

Since both are riding the train together, the number of rides will stay the same. We can add the money amounts and keep the x the same in the simplified expression.

.85 + .60 = 1.45

Our answer is 1.45x .

If Kara multiplies the number of train rides that the three of them take times $1.45 then she will have the total amount of money needed or spent riding the train.

Vocabulary

Expression
a number sentence without an equal sign that combines numbers, variables and operations.
Simplify
to make smaller by combining like terms.
Sum
the answer in an addition problem.
Difference
the answer in a subtraction problem.
Product
the answer in a multiplication problem.
Quotient
the answer in a division problem.

Guided Practice

Here is one for you to try on your own.

Samera has twice as many pets as Amit has. Kyra has 4 times as many pets as Amit has. Let a represent the number of pets Amit has.

a. Write an expression to represent the number of pets Samera has.

b. Write an expression to represent the number of pets Kyra has.

c. Write an expression to represent the number of pets Samera and Kyra have all together.

Answer

Consider part a first.

The phrase “twice as many pets as Amit” shows how many pets Samera has. Use a number, an operation sign, or a variable to represent each part of that phrase.

& \underline{twice} \ as \ many \ pets \ as \ \underline{Amit}\\& \downarrow \qquad \qquad \qquad \qquad \qquad \downarrow\\& 2 \times \qquad \qquad \qquad \quad \qquad a

So, the expression 2 \times a or 2a represents the number of pets Samera has.

Consider part b next.

The phrase “4 times as many pets as Amit” shows how many pets Kyra has. Use a number, an operation sign, or a variable to represent each part of that phrase.

& \underline{4} \ \underline{times} \ as \ many \ pets \ as \ Amit\\& \downarrow \quad \downarrow \qquad \qquad \qquad \qquad \downarrow\\& 4 \quad \times \qquad \qquad \qquad \quad \ \ a

So, the expression 4 \times a or 4a represents the number of pets Kyra has.

Finally, consider part c .

To find the number of pets Samera and Kyra have “all together,” write an addition expression.

& (\text{number of pets Samera has}) \ + \ (\text{number of pets Kyra has})\\& \qquad \qquad \downarrow \qquad \qquad \qquad \qquad \ \ \downarrow \qquad \qquad \quad \quad \ \downarrow\\& \qquad \qquad 2a \qquad \qquad \qquad \quad  \ \ + \qquad \qquad  \quad \quad 4a

Simplify the expression.

2a+4a=6a

The number of pets Samera and Kyra have all together can be represented by the expression 6a .

Video Review

- This is a James Sousa video about combining like terms to simplify an expression.

Practice

Directions: Simplify each variable expression involving multiple operations.

1. 6a+4a-2b

2. 16b-4b \cdot 2

3. 22a \div 2+14a

4. 19x-5x \cdot 2

5. 16y-12y \div 2

6. 16a-4a-12b

7. 26a+14a+12b+2b

8. 36a+4a-2b+5b

9. 18a+4a+12y

10. 46a+34a-12b+14b

11. 16y+4y-2x

12. 6x+4x+2x+4y-19z

13. 26y-12y \div 2

14. 36y-12y \div 12

15. 46y+12y \div 2

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