Lesson 9

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 \(7a+\text-2a+4\).
4. \(8a-(8a-8)\) is equivalent to 8.

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

• 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.
   \(\displaystyle 7a+5b-3a+4b\) \(\displaystyle (a+a+a+a+a+a+a) + (b+b+b+b+b) - (a+a+a) + (b+b+b+b)\) \(\displaystyle (a+a+a+a) + (a+a+a) + (b+b+b+b+b) - (a+a+a) + (b+b+b+b)\) \(\displaystyle (a+a+a+a) + (b+b+b+b+b) + (a+a+a) - (a+a+a) + (b+b+b+b)\) \(\displaystyle (a+a+a+a) + (b+b+b+b+b) + (b+b+b+b)\) \(\displaystyle (a+a+a+a) + (b+b+b+b+b+b+b+b+b)\) \(\displaystyle 4a + 9b\)

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. \(\displaystyle 7a+5b-3a+4b\) \(\displaystyle 7a+5b+(\text-3a)+4b\) \(\displaystyle 7a+(\text-3a)+5b+4b\) \(\displaystyle (7+\text-3)a+(5+4)b\) \(\displaystyle 4a+9b\)

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. \(6x+{?}=\text-10x\)
   
   
    
4. \(6x+{?}=0\)
   
   
    
5. \(6x+{?}=10\)
   
   
    

Check your results with your partner and resolve any disagreements. Next move on to Set B.

Set B

1. \(6x - {?}= 2x\)
   
   
    
2. \(6x - {?} = 10x\)
   
   
    
3. \(6x - {?} = x\)
   
   
    
4. \(6x - {?} = 6\)
   
   
    
5. \(6x - {?} = 4x-10\)
   
   
    

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, \(2(\text-3+x)+8\) 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, \(2(\text-3+x)+8\) is equivalent to \(2x+2\) because:

\(\begin{align} 2(\text-3+x)+8\\ \text-6+2x+8 & \quad\text{by the distributive property}\\ 2x+\text-6+8 & \quad\text{by the commutative property}\\ 2x+(\text-6+8) & \quad\text{by the associative property} \\ 2x+2 \\ \end{align}\)