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AProve that for all integers x and $y, { }^{-}x - y = { }^{-}y - x$.
BDoes part (a) imply that subtraction is commutative? Explain.



What 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 :

  • Prove \(-x - y = -y - x\) using the definition of subtraction as the addition of an additive inverse and the commutative property of addition.
  • Subtraction is not commutative because \(x - y eq y - x\) for arbitrary integers.
  • The first step in the proof is changing the equation into an addition expression (Option B).
  • The expression \(-x - y\) becomes \(-x + (-y)\) (missing term is \(-y\)).
  • The rule used for this transformation is \(a - b = a + (-b)\) (Option D).

Explanation

  1. Proof of the Identity \(-x - y = -y - x\)

To analyze the expression \(-x - y = -y - x\) for all integers \(x\) and \(y\), we use the definition of subtraction. The subtraction of an integer is defined as the addition of its additive inverse: \(a - b = a + (-b)\). Applying this definition to the left side: \(-x - y = (-x) + (-y)\). Similarly, for the right side: \(-y - x = (-y) + (-x)\). Because addition of integers is commutative, we know that \((-x) + (-y) = (-y) + (-x)\). Therefore, \(-x - y = -y - x\).

  1. Analysis of Commutativity of Subtraction

Part (a) proves that \(-x - y = -y - x\), which means we can swap the order of the terms if both terms are being subtracted. However, this does not imply that subtraction itself is commutative. An operation is commutative if \(a - b = b - a\) for all \(a, b\). If we test this with values, such as \(x = 5\) and \(y = 3\), we find \(5 - 3 = 2\) but \(3 - 5 = -2\). Since \(2 eq -2\), subtraction is not commutative. The equality \(-x - y = -y - x\) is a specific property of negations and does not generalize to the subtraction of any two arbitrary integers.

  1. Step Selection for the Proof

To prove the identity \(-x - y = -y - x\), it is most helpful to convert the subtraction operations into addition. By writing the expressions as sums of additive inverses, we can then apply the well-known commutative property of addition. Thus, the first step should be to change both sides of the equation into an addition expression. This corresponds to option B.

  1. Rewriting the Expression as Addition

To rewrite the left side of the equation \(-x - y\) as an addition expression, we use the definition \(a - b = a + (-b)\). Replacing \(a\) with \(-x\) and \(b\) with \(y\), we get: \(-x - y = -x + (-y)\). Therefore, the missing term is \(-y\).

  1. Identifying the Theorem Used

The theorem used to justify the transformation \(-x - y = -x + (-y)\) is the definition of subtraction as the addition of an additive inverse. This is stated as: for all \(a\) and \(b\), \(a - b = a + (-b)\). This corresponds to option D.

Examples

Understanding the properties of subtraction is useful in balancing budgets or tracking inventory. For instance, if you have a starting balance of \(\$100\) and you make two consecutive withdrawals of \(\$20\) and \(\$30\), the order of withdrawals doesn't change the final balance: \(100 - 20 - 30 = 50\) and \(100 - 30 - 20 = 50\). However, swapping the order of the amounts themselves (i.e., \(20 - 30\) instead of \(30 - 20\)) would lead to different results, highlighting that while consecutive subtractions are commutative, the operation of subtraction itself is not. Mastering these identities helps prevent errors when manipulating financial data or organizing resources.

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