Answered

AWhat first step should be taken in proving ${ }^{-} x - y ={ }^{-} y - x$ for all integers $x$ and $y$?
ASubtract $x + y$ from both sides of the equation.
BChange both sides of the equation to an addition expression.
CSubtract $y$ from both sides of the equation.
DSubtract $x$ from both sides of the equation.



Change the left side of the equation to an addition expression.

${ }^{-} x-y={ }^{-} x+$



What theorem is used in the previous step?

AFor all $a$, ${ }^{-} ext{ }(^{-} a) = a$.
BFor all $a$, $a + { }^{-} a = 0 = { }^{-} a + a$.
CFor all $a$ and $b$, ${ }^{-} a + { }^{-} b = { }^{-}(a + b)$.
DFor all $a$ and $b$, $a - b = a + { }^{-} b$.

Verified Answer :

  • The first step to proving the identity is changing the expression \(-x - y\) into addition, resulting in \(-x + (-y)\).
  • This step is justified by the theorem \(a - b = a + (-b)\).
  • By the commutative property, \(-x + (-y) = -y + (-x)\), which equals \(-y - x\).
  • The correct selections are: First step B, blank is \((-y)\), and Theorem is D.

Explanation

  1. Analyzing the Initial Step and Definition

To prove the identity \(-x - y = -y - x\) for all integers \(x\) and \(y\), we start by transforming the left-hand side of the equation. The left-hand side is \(-x - y\). According to the fundamental definition of subtraction in integer arithmetic, subtracting an integer is equivalent to adding its additive inverse. This is stated as the theorem: for all integers \(a\) and \(b\), \(a - b = a + (-b)\).

  1. Executing the Transformation and Identifying the Theorem

Applying the theorem \(a - b = a + (-b)\) to the expression \(-x - y\), where \(a = -x\) and \(b = y\), we obtain:
\[-x - y = -x + (-y)\]
This transformation allows us to change the subtraction into an addition expression. Therefore, the blank in the expression \(-x - y = -x + \boxed{\phantom{xxx}}\) should be filled with \((-y)\). Comparing this to the given choices, "Change both sides of the equation to an addition expression" (Option B) describes the first step of this proof, and the theorem used to justify this transformation is option D: "For all \(a\) and \(b\), \(a - b = a + (-b)\)."

  1. Conclusion of the Proof and Summary

To complete the proof, we use the commutative property of addition, which states that \(a + b = b + a\).
Applying this to our expression:
\[-x + (-y) = -y + (-x)\]
Finally, applying the definition of subtraction in reverse (where \(a + (-b) = a - b\)), we get:
\[-y + (-x) = -y - x\]
Thus, we have shown \(-x - y = -y - x\). The initial step and the theorem used are identified as follows:

  • First step: Change the left side of the equation to an addition expression (Option B).
  • Blank fill: \(-y\).
  • Theorem: Option D, which states \(a - b = a + (-b)\).

Examples

This proof demonstrates how fundamental algebraic properties, such as the definition of subtraction and the commutative property of addition, are used to manipulate and verify mathematical statements. In real life, these principles are used in computer science for designing efficient algorithms, in financial modeling for balancing accounts, and in engineering for ensuring that physical systems obey consistent rules of transformation. Just as we proved \(-x - y = -y - x\), these logical foundations allow us to simplify complex expressions into manageable parts, ensuring accuracy in every calculation.