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

4/7k+(j+1/7k)

To identify the property or rule that justifies each step in the calculation, we need to break down the calculation step by step:

Step 1: 4/7k + (j + 1/7k)

In this step, we have an expression with two terms inside parentheses. To simplify this expression, we need to follow the order of operations, which states that we should first perform any calculations inside the parentheses.

Step 2: 4/7k + j + 1/7k

In this step, the property that justifies the addition of the terms within parentheses is the Associative Property of Addition. According to this property, changing the grouping of the terms does not change the sum. So, we can change the grouping of (j + 1/7k) to j + (1/7k).

Step 3: 4/7k + 1/7k + j

In this step, we added the fractions 4/7k and 1/7k. The property that justifies this step is the Commutative Property of Addition, which states that the order of the terms can be changed without affecting the sum.

Step 4: (4 + 1)/7k + j

In this step, we simplified the sum of the fractions 4/7k and 1/7k by adding the numerators together. The property that justifies this step is the Additive Property of Fractions, which states that to add fractions with the same denominator, we add the numerators and keep the common denominator.

Step 5: 5/7k + j

In this step, we have simplified the expression by adding the fractions with the same denominator. The property that justifies this step is the Simplification Property of Addition, which states that we can simplify an expression by combining like terms, i.e., terms with the same variables raised to the same powers.

Therefore, the properties or rules that justify each step in the calculation are the Associative Property of Addition, the Commutative Property of Addition, the Additive Property of Fractions, and the Simplification Property of Addition.