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Contributor: Ashley Nail. Lesson ID: 13900
Everyone got different homework answers, but they're all correct! How is that possible? Use what you know about combining like terms and the distributive property to identify equivalent expressions.
Ailee, Liam, and Maddy are working on their homework together.
On one problem, they all got these different answers.
| Ailee | Liam | Maddy | |
| 4x | 3x + x | 2(2x) |
Their teacher tells them that each of their answers is correct.
Ailee has the idea to plug a number value into each expression.
She replaces the variable x with 10.
| Ailee | Liam | Maddy | |
| 4x | 3x + x | 2(2x) | |
| 4 • 10 | 3 • 10 + 10 | 2(2 • 10) | |
| 40 | 30 + 10 | 2 • 20 | |
| 40 | 40 |
All three expressions equal 40 when x = 10.
Maddy still doesn’t believe the expressions are equivalent.
She wants to plug 9 into each expression.
| Ailee | Liam | Maddy | |
| 4x | 3x + x | 2(2x) | |
| 4 • 9 | 3 • 9 + 9 | 2(2 • 9) | |
| 36 | 27 + 9 | 2 • 18 | |
| 36 | 36 |
Again, the expressions all equal the same value when x = 9.
Liam says you can prove the expressions are equivalent without even replacing the variable with a number. See if he’s correct.
First, look at this expression.
3x + x
You can rewrite the expression as the following.
3x + 1x
Since both terms contain the variable x, they can be added together.
3x + 1x = 4x
Next, look at this expression.
2(2x)
You can use the distributive property to multiply everything inside the parentheses by the factor outside of the parentheses.
You can rewrite the expression as follows.
2 • 2x
Now, you multiply.
2 • 2x = 4x
So all three students’ expressions are equivalent. No matter what number replaces the variable, the value of the expressions will all be equal.
Before you practice on your own, look at one more example.
3(x + 5y + 3)
18xy + 9
3x + 15y + 9
3(x + y + 9)
First, see if you can simplify the original expression.
Two terms share the same variable x so they can be combined.
You cannot combine the x and y terms.
This shows you right away one equivalent expression and eliminates one expression from the list.
3(x + 5y + 3)
18xy + 9
3x + 15y + 9 √
3(x + y + 9)
Now, use the distributive property to factor a number out of each term.
Each term shares a factor of 3.
You can factor 3 out of each term and place it outside the parentheses. This shows you the last equivalent expression!
3(x + 5y + 3) √
18xy + 9
3x + 15y + 9 √
3(x + y + 9)
Now you are ready to practice finding equivalent expressions on your own!
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