We are going to break down each step and see whether is it associative, communicative, or distributive, or property of equality for addition or multiplication.
Given:
5( x + 3 ) + 2x = 4x + 9
Step 1:
5x + 15 + 2x = 4x + 9
Here, we can see that 5 from 5( x + 3 ) has been distributed to the terms inside the parenthesis, making it 5x + 15. This step is done through Distributive Property of Addition.
Step 2:
5x + 2x + 15 = 4x + 9
In here, 2x and 15 have switched places. According to the Commutative Property of Addition, a + b = b + a, applying this to our given, 15 + 2x = 2x + 15.
Step 3:
( 5 + 2 )x + 15 = 4x + 9
7x + 15 = 4x + 9
Here, we can see that x has been isolated from 5x + 2x, giving ( 5 + 2 )x. If we go back to our Distributive Property, we can see that this is just a reverse of that. Once we distribute x to ( 5 + 2 ), we will get 5x + 2x. Hence, this is also done through the Distributive Property of Addition.
Step 4:
7x + 15 - 15 = 4x + 9 - 15
7x = 4x - 6
In this part, we can see that 15 was subtracted from both sides of the equation. We can also say the -15 was added to both sides of the equation. If we go back to the definition of the Addition Property of Equality, it states that if two expressions are equal to each other, and you add the same value to both sides of the equation, the equation will remain equal. Since we added -15 to both sides of the equation, this step is done through the Addition Property of Equality.
Step 5:
7x - 4x = 4x - 6 - 4x
Here, just like the 4th step, we added -4x to both sides of the equation, hence this step is also done through Addition Property of Equality.
Step 6:
( 7 - 4 )x = -6
3x = -6
What we see here is quite similar to the