Of course! Let's carefully break down the steps to identify the properties used in the given expressions:
Given expressions:
[tex]\[ 10 \cdot(9 + 2 \cdot x) = 10 \cdot 9 + 10 \cdot 2 \cdot x = 20x + 90 \][/tex]
1. Step 1: Applying the Distributive Property
The first part of the expression involves the distributive property, which states that:
[tex]\[ a \cdot (b + c) = a \cdot b + a \cdot c \][/tex]
Here, the expression:
[tex]\[ 10 \cdot (9 + 2 \cdot x) \][/tex]
can be expanded using the distributive property:
[tex]\[ 10 \cdot 9 + 10 \cdot 2 \cdot x \][/tex]
2. Step 2: Simplifying the Terms
Now we simplify the terms:
[tex]\[ 10 \cdot 9 = 90 \][/tex]
[tex]\[ 10 \cdot 2 \cdot x = 20x \][/tex]
So we have:
[tex]\[ 90 + 20x \][/tex]
3. Step 3: Utilizing the Commutative Property of Addition
Finally, the expression can be rearranged using the commutative property of addition, which states that:
[tex]\[ a + b = b + a \][/tex]
Therefore:
[tex]\[ 90 + 20x \][/tex]
is equivalent to:
[tex]\[ 20x + 90 \][/tex]
Conclusion:
The two properties that justify equating the expressions are:
1. Distributive Property
2. Commutative Property of Addition
These properties explain how the initial expression is expanded and rearranged to reach the final form.
