Illustrative Mathematics Grade 7, Unit 6, Lesson 20: Combining Like Terms

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Illustrative Math
Grade 7

Lesson 20: Combining Like Terms

Let’s see how we can tell that expressions are equivalent.

Illustrative Math Unit 7.6, Lesson 20 (printable worksheets)

Lesson 20 Summary

There are many ways to write equivalent expressions that may look very different from each other. We have several tools to find out if two expressions are equivalent.

• Two expressions are definitely not equivalent if they have different values when we substitute the same number for the variable. For example, 2(-3 + x) + 8 and 2x + 5 are not equivalent because when x is 1, the first expression equals 4 and the second expression equals 7.
• If two expressions are equal for many different values we substitute for the variable, then the expressions may be equivalent, but we don’t know for sure. It is impossible to compare the two expressions for all values. To know for sure, we use properties of operations. For example, 2(-3 + x) + 8 is equivalent to 2x + 2 because:
  2(-3 + x) + 8
  -6 + 2x + 8 by the distributive property
  2x + -6 + 8 by the commutative property
  2x +2 by the associative property

Lesson 20.1 Why is it True?

Explain why each statement is true.

1. 5 + 2 + 3 = 5 + (2 + 3)
2. 9a is equivalent to 11a - 2a.
3. 7a + 4 - 2a is equivalent to 7a + -2a + 4.
4. 8a -(8a - 8) is equivalent to 8.

• See Video for Whole Lesson

Lesson 20.2 A’s and B’s

Diego and Jada are both trying to write an expression with fewer terms that is equivalent to
7a + 5b - 3a + 4b

• Jada thinks 10a + 1b is equivalent to the original expression.
• Diego thinks 4a + 9b is equivalent to the original expression.
  

1. We can show expressions are equivalent by writing out all the variables. Explain why the expression on each row (after the first row) is equivalent to the expression on the row before it.
   7a + 5b - 3a + 4b
   (a+a+a+a+a+a+a)+(b+b+b+b+b)-(a+a+a)+(b+b+b+b)
   (a+a+a+a)+(a+a+a)+(b+b+b+b+b)-(a+a+a)+(b+b+b+b)
   (a+a+a+a)+(a+a+a)-(a+a+a)+(b+b+b+b+b)+(b+b+b+b)
   (a+a+a+a)+(b+b+b+b+b)+(b+b+b+b)
   (a+a+a+a)+(b+b+b+b+b+b+b+b+b)
   4a + 9b
   
2. Here is another way we can rewrite the expressions. Explain why the expression on each row (after the first row) is equivalent to the expression on the row before it.
   7a + 5b - 3a + 4b
   7a + 5b + (-3a) + 4b
   7a + (-3a) + 5b + 4b
   (7 + -3)a + (5 + 4)b
   4a + 9b
   

Are you ready for more?

Follow the instructions for a number puzzle:

• Take the number formed by the first 3 digits of your phone number and multiply it by 40
• Add 1 to the result
• Multiply by 500
• Add the number formed by the last 4 digits of your phone number, and then add it again
• Subtract 500
• Multiply by

1. What is the final number?
   • Show Answer

     The final number is the initial 7-digit phone number.

2. How does this number puzzle work?
   • Show Answer

     Let x be the first 3 digits and y be the last 4 digits.
     Then we can get the following expression from the steps of the puzzle.
     ((500(40x + 1) + y + y) - 500)1/2
     = 1/2((500(40x + 1) + 2y)- 500)
     = 1/2(20000x + 500 + 2y - 500)
     = 10000x + y
     

3. Can you invent a new number puzzle that gives a surprising result?

Lesson 20.3 Making Sides Equal

Replace each ? with an expression that will make the left side of the equation equivalent to the right side.
Set A

1. 6x + ? = 10x
2. 6x + ? = 2x
3. 6x + ? = -10x
4. 6x + ? = 0
5. 6x + ? = 10 Check your results with your partner and resolve any disagreements. Then move on to Set B.
   Set B
   
6. 6x - ? = 2x
7. 6x - ? = 10x
8. 6x - ? = x
9. 6x - ? = 6
10. 6x - ? = 4x - 10

Lesson 20 Practice Problems

1. Andre says that 10x + 6 and 5x + 11 are equivalent because they both equal 16 when is 1. Do you agree with Andre? Explain your reasoning.
2. Select all expressions that can be subtracted from 9x to result in the expression 3x + 5.
3. Select all the statements that are true for any value of x.
4. For each situation, would you describe it with x < 25, x > 25, x ≤ 25, or x ≥ 25?
   a. The library is having a party for any student who read at least 25 books over the summer. Priya read x books and was invited to the party.
   b. Kiran read x books over the summer but was not invited to the party.
   
5. Consider the problem: A water bucket is being filled with water from a water faucet at a constant rate. When will the bucket be full? What information would you need to be able to solve the problem?

The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.

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