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Lesson 6
Module 3
equivalent expressions
equivalent expressions
Liam inserted his card into the terminal, typed in his PIN number, then hit the cash back button. He wanted to get $20 out of his account. Automatically, the clerk replied, "How would you like it back?" meaning, how would you like it to be broken down? There were many combinations that $20 could be broken into. The clerk could have given him:
- one twenty-dollar bill
- two ten-dollar bills
- four five-dollar bills
- one ten-dollar bill, one five-dollar bill, and five one-dollar bills
Objectives: - understand that an expression written in different ways can be equivalent
- identify when two expressions are eqiuvalent
Goals
The table shows the total cost for each family, but what about an expression that is equivalent to the sum of both individual expressions? Can there be an expression for that? Look at these two expressions below:
Equivalent EXpressions-scenario
Questions:
Do these two expressions represent the sum correctly? Are these two expressions equivalent? Can you prove it using mathematical operations and properties?
The McMillen and Dunlap families went to the same carnival over the weekend. Each family purchased amusement tickets and popcorn combos. Let the variable t represent the price per ticket and p represent the price per popcorn combos. Here is the breakdown of their expenses:
Parts of an Expression
check your work
exploration
The McMillen family’s cost is represented with the expression 8t + 6p. The Dunlap family’s cost is represented with the expression 10t + 8p. Notice how both expressions contain two terms. You would think if a two-term expression was added to another two-term expression, there would be four terms total, right? But why does the sum of these two expressions result in a two-term expression, 18t + 14p? This could only happen if some of the terms were combined. Take, for example, 4 + 3 = 7. One number plus one number results in one number, not two separate numbers. This is an example of combining like terms! What are like terms, and how do they work with an algebraic expression? Let’s take a look. Be sure to review the parts of an express if you need a quick reminder.
Please complete to check your understanding
Practice
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like terms
unlike terms
exploration
Sometimes, an expression can contain terms that have the same variable. These terms are called like terms. Like terms have identical variable parts but may have different coefficients. When terms have variable parts that are different, they are called unlike terms. Let’s look at a few examples of like and unlike terms:
Combining like terms
combine like terms
Combining like terms means combining the coefficients of the like terms. The purpose of combining like terms is to help simplify an expression. Take a look at the examples below to understand what it means to combine like terms:
click for answer
click for answer
Complete these problems on your own before checking your answer.
3x + 5x
Combine the like terms in the expression shown below. Be sure to show all your steps.
7t + 10 + 3
Combine the like terms in the expression shown below. Be sure to show all your steps.
practice:
Use the mathematical properties to prove that 2(x + 5 + 7x) is equivalent to 15x + 10.
Use the mathematical properties to prove that 8b + 0 − b is equivalent to 8b.
Show that the expression 4x + 1 + 3x + 8 is equivalent to 7x + 9. Can you identify which properties are used to show that the two expressions are equivalent? Think about which property allows you to move from step to step.
Example 3:
Example 2:
Example 1:
Show with substitution
Show how they are equivalent
Show with another number
MATHEMATICal PROPERTIES
In an algebraic expression, you can use the mathematical properties to group and combine like terms. The properties also allow you to prove that two expressions are equivalent. Take a look at the examples below to learn how to identify if other expressions are equivalent using various properties. Be sure to review the mathematical properties if needed, as you will use them to identify equivalent expressions.
Show with substitution
Show with substitution
Show how they are equivalent
Show with another number
mathematical properties
False
True ✓
Question 1/5
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learning session / 02
correct answer
His audiam deserunt in, eum ubique voluptatibus te. In reque dicta usu. Ne rebum dissentiet eam, vim omnis deseruisse id. Ullum deleniti vituperata at quo.
Write the wrong option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
Write the correct option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
Write the wrong option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
learning session / 02
Question 2/5
Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua?
correct answer
His audiam deserunt in, eum ubique voluptatibus te. In reque dicta usu. Ne rebum dissentiet eam, vim omnis deseruisse id. Ullum deleniti vituperata at quo.
learning session / 02
Question 3/5
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Write the wrong option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
Write the correct option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
Write the wrong option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
correct answer
His audiam deserunt in, eum ubique voluptatibus te. In reque dicta usu. Ne rebum dissentiet eam, vim omnis deseruisse id. Ullum deleniti vituperata at quo.
Write the wrong option here
learning session / 02
Question 4/5
Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua?
Write the wrong option here
Write the correct option here
Write the wrong option here
correct answer
His audiam deserunt in, eum ubique voluptatibus te. In reque dicta usu. Ne rebum dissentiet eam, vim omnis deseruisse id. Ullum deleniti vituperata at quo.
Write the wrong option here
Write the wrong option here
Write the wrong option here
learning session / 02
Question 5/5
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Write the wrong option here
Write the correct option here
Write the wrong option here
correct answer
His audiam deserunt in, eum ubique voluptatibus te. In reque dicta usu. Ne rebum dissentiet eam, vim omnis deseruisse id. Ullum deleniti vituperata at quo.
learning session / 02
Summary
- Lorem ipsum dolor sit amet consectetur adipiscing, elit nunc taciti luctus consequat ultricies turpis,
- Lobortis nulla vehicula ultrices egestas. Eu metus quam luctus rutrum eleifend ornare egestas dictumst velit.
- Vulputate mauris lacinia commodo diam dignissim nisi tempor quis integer fermentum lobortis imperdiet.
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LOREM IPSUM DOLOR SIT AMER
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Nunc lobortis dui in feugiat fames mattis sodales
Nunc lobortis dui in feugiat fames mattis sodales
Nunc lobortis dui in feugiat fames mattis sodales
Nunc lobortis dui in feugiat fames mattis sodales
Nunc lobortis dui in feugiat fames mattis sodales
Nunc lobortis dui in feugiat fames mattis sodales
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learning session / 03
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Lorem ipsum dolor sit amet consectetur adipiscing elit
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case study resolution
example
example
example
A process of combining terms that have identical variable parts.
Terms in an expression that do not have the same exact variable part; for example, 5x and y are unlike terms in the expression 5x + y.
combine like terms
Unlike terms
Like terms
Terms in an expression that have the same exact variable part; for example, 5x and 4x are like terms in the expression 5x + 4x.
vocabulary
In an algebraic expression, you can use the mathematical properties to group and combine like terms. The properties also allow you to prove that the simplified expression is equivalent to the beginning expression. Example: Prove the expression 8t + 6p + 10t + 8p is equivalent to the expression 18t + 14p.
Proving Expressions Are Equivalent
Combining like terms means to combine the coefficients of the like terms. Remember, the coefficient of a variable includes the number and the sign in front of the number. The purpose of combining like terms is to simplify an expression. Simplifying makes it easier to evaluate the expression and to prove if it is equivalent to another expression.
Combining Like Terms
Proof using mathematical properties and simplifying:8t + 6p + 10t + 8p = 8t + 10t + 6p + 8p Use the commutative property. = (8 + 10)t + (6 + 8)p Use the distributive property. = 18t + 14p Combine like terms.= 18t + 14p Proof using substitution: 18t + 14p 8t + 6p + 10t + 8p = 18(1) + 14(1) 8(1) + 6(1) + 10(1) + 8(1) = 18 + 14 8 + 6 + 10 + 8 = 32 32
Proof Expressions are Equivalent
Like terms have identical variables but may have different coefficients. When terms have vari ables that are different they are called unlike terms.
Like and Unlike Terms
summary
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Self-evaluation
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1/5
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Write the wrong option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
Write the correct option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
Write the wrong option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
correct answer
His audiam deserunt in, eum ubique voluptatibus te. In reque dicta usu. Ne rebum dissentiet eam, vim omnis deseruisse id. Ullum deleniti vituperata at quo.
2/5
Write the wrong option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
Write the wrong option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
Write the correct option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua?
correct answer
His audiam deserunt in, eum ubique voluptatibus te. In reque dicta usu. Ne rebum dissentiet eam, vim omnis deseruisse id. Ullum deleniti vituperata at quo.
3/5
Write the wrong option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
Write the correct option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
Write the wrong option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
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correct answer
His audiam deserunt in, eum ubique voluptatibus te. In reque dicta usu. Ne rebum dissentiet eam, vim omnis deseruisse id. Ullum deleniti vituperata at quo.
4/5
Write the wrong option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
Write the correct option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
Write the wrong option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua?
correct answer
His audiam deserunt in, eum ubique voluptatibus te. In reque dicta usu. Ne rebum dissentiet eam, vim omnis deseruisse id. Ullum deleniti vituperata at quo.
5/5
Write the wrong option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
Write the correct option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
Write the wrong option here Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
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correct answer
His audiam deserunt in, eum ubique voluptatibus te. In reque dicta usu. Ne rebum dissentiet eam, vim omnis deseruisse id. Ullum deleniti vituperata at quo.
Summary
- Lorem ipsum dolor sit amet consectetur adipiscing, elit nunc taciti luctus consequat ultricies turpis,
- Lobortis nulla vehicula ultrices egestas. Eu metus quam luctus rutrum eleifend ornare egestas dictumst velit.
- Vulputate mauris lacinia commodo diam dignissim nisi tempor quis integer fermentum lobortis imperdiet.
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NO! 4x² and 2x are not like terms because the variable parts are not identical. One variable is squared, and the other is not.
4x and x are like terms in this algebraic expression. In addition, the constants 2 and 9 are like terms in this expression.
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YES! 6x and -5x are like terms because the variable parts are identical.
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NO! 2x and 2p are unlike terms because the variable parts are not identical.
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