Final answer:
The transition from Line 1 to Line 2 uses the Commutative Property of Addition. Line 3 simplifies the expression using the Distributive Property of multiplication over addition.
Step-by-step explanation:
The transition from Line 1 to Line 2 in the given mathematical expression uses a property that allows the order of the terms being added inside the parentheses to be changed without affecting the value of the expression. This property is known as the Commutative Property, specifically the Commutative Property of Addition, which states that the order in which two numbers are added does not change the sum, i.e., a + b = b + a. Thus, in moving from Line 1: 4(5+6x) to Line 2: 4(6x+5), the Commutative Property justifies the rearrangement of the terms 5 and 6x within the parentheses.
Line 3, which simplifies to 24x+20, involves the use of the Distributive Property of multiplication over addition. This property states that multiplying a sum by a number yields the same result as multiplying each addend by the number and then adding the products, expressed as a(b+c) = ab + ac. Here, the number 4 is distributed over the terms inside the parentheses, resulting in 24x+20.