prove -x-y=-y-x, for all integers
I do not know what grade you're in, but questions of this sort could go quite deep into algebra.
Since x,y are integers, which belong to a ring in which additions are commutative, so if we write the subtractions as inverses of addition, the above would be invoking commutativity in the ring-addition:
-x-y
=(-x)+(-y) inverse of addition
=(-y)+(-x) commutativity of addition
=-y-x inverse of addition
For high-school algebra, then it would be the commutative property of addition and subtraction.
To prove that -x-y=-y-x for all integers, we need to use the properties of integer addition and the definition of negative numbers.
1. Start with the left-hand side of the equation: -x-y.
2. According to the definition of negative numbers, -x is the additive inverse of x. This means that -x + x = 0 (the sum of x and its additive inverse is zero).
3. Applying this property to the expression -x-y, we can rewrite it as (-x + x) - y.
4. Since the addition of real numbers is associative, we can rearrange the expression as -x + (x - y).
5. Now, let's focus on the term (x - y). By the commutative property of addition, we can rewrite it as (y - x).
6. Substituting this back into our expression, we have -x + (y - x).
7. According to the associative property of addition, we can rearrange the expression as (y - x) - x.
8. Finally, by the commutative property of addition, we get y - (x + x).
9. Simplifying further, we have y - 2x.
10. Now, let's compare this with the right-hand side of the equation, -y-x.
11. By the commutative property of addition, we can rearrange it as -y - x.
12. Now, if we compare this with step 10, we can see that they are the same expression.
13. Therefore, we have proven that -x-y = -y-x holds true for all integers.
In conclusion, we proved the equality -x-y=-y-x by utilizing the properties of integer addition, such as the associative and commutative properties, and the definition of negative numbers.