Certainly! Let's analyze each statement one by one to identify the properties of equality used.
Statement 1: If [tex]\( 6 + 11 = 17 \)[/tex], then [tex]\( 17 = 6 + 11 \)[/tex].
To understand the property of equality at work here, let's break it down:
- We know that [tex]\( 6 + 11 = 17 \)[/tex].
- In the second part, [tex]\( 17 = 6 + 11 \)[/tex], the left-hand side (17) is now the right-hand side, and the right-hand side ([tex]\( 6 + 11 \)[/tex]) is now the left-hand side.
- This implies that if one quantity equals another, reversing the equation also holds true.
The property of equality that justifies this statement is the Symmetric Property of Equality, which states:
[tex]\[ \text{If } a = b, \text{ then } b = a. \][/tex]
Thus, for statement 1, the property used is the Symmetric Property of Equality.
Statement 2: [tex]\( z + 6 = z + 6 \)[/tex].
To understand the property at play here, consider the following:
- An equation of the form [tex]\( z + 6 = z + 6 \)[/tex] means that a quantity is being equated to itself.
- There is no change to either side; it strictly confirms that any quantity is equal to itself.
The property of equality that justifies this statement is the Reflexive Property of Equality, which states:
[tex]\[ a = a. \][/tex]
Thus, for statement 2, the property used is the Reflexive Property of Equality.
Summary:
1. For the statement "If [tex]\( 6 + 11 = 17 \)[/tex], then [tex]\( 17 = 6 + 11 \)[/tex]":
- The property used is the Symmetric Property of Equality.
2. For the statement "[tex]\( z + 6 = z + 6 \)[/tex]":
- The property used is the Reflexive Property of Equality.
Both properties are integral to understanding the fundamental principles of equality in mathematics.
