Identify the property or rule that justifies each step in the calculation.

4/7k+(j+1/7k) ____________
4/7k+(1/7k+j) ___________
(4/7k+1/7k)+j _____________
k(4/7+1/7)+j _________________
5/7k+j ___________

addition associative

addition commutative
add assoc
distributive of mult over add
mult comm

To identify the property or rule that justifies each step in the calculation, let's go through each step one by one.

Step 1: 4/7k + (j + 1/7k)
In this step, the rule used is the Associative Property of Addition. This property states that for any real numbers a, b, and c, the sum of a and the sum of b and c will be the same regardless of how we group them. Therefore, we can group (j + 1/7k) together without changing the overall value of the expression.

Step 2: 4/7k + (1/7k + j)
Here, the Commutative Property of Addition is used. This property states that the order of the numbers being added does not affect the result. As a result, we can switch the order of the terms (j + 1/7k) to (1/7k + j) without changing the value.

Step 3: (4/7k + 1/7k) + j
In this step, the rule followed is the Associative Property of Addition again. By grouping the terms (4/7k + 1/7k) together, we preserve the value of the expression. This property allows us to add the two fractions (4/7k) and (1/7k) first before adding the result to j.

Step 4: k(4/7 + 1/7) + j
Here, the Distributive Property is applied. The Distributive Property states that multiplying a number by the sum of two other numbers is the same as multiplying the number by each of the two numbers separately and then adding the results. Therefore, we can distribute the factor k to both (4/7) and (1/7) individually before adding the result to j.

Step 5: 5/7k + j
In the final step, we simplify the expression. By adding (4/7) and (1/7), we get (5/7). Therefore, the final expression is 5/7k + j.

Overall, the properties and rules used in the calculation are the Associative Property of Addition, the Commutative Property of Addition, and the Distributive Property.