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Answer:
(a) -(x+7)/y
(b) (x+7)/-y
Step-by-step explanation:
There are several ways you can show expressions are equivalent. Perhaps the easiest and best is to put them in the same form. For an expression such as this, I prefer the form of answer (a), where the minus sign is factored out and the numerator and denominator have positive coefficients.
The given expression with -1 factored out is ...
![\dfrac{-x-7}{y}=\dfrac{1(x+7)}{y}=\boxed{-\dfrac{x+7}{y}} \quad\text{matches A}](https://tex.z-dn.net/?f=%5Cdfrac%7B-x-7%7D%7By%7D%3D%5Cdfrac%7B1%28x%2B7%29%7D%7By%7D%3D%5Cboxed%7B-%5Cdfrac%7Bx%2B7%7D%7By%7D%7D%20%5Cquad%5Ctext%7Bmatches%20A%7D)
Likewise, the expression of (b) with the minus sign factored out is ...
![\dfrac{x+7}{-y}=\boxed{-\dfrac{x+7}{y}}](https://tex.z-dn.net/?f=%5Cdfrac%7Bx%2B7%7D%7B-y%7D%3D%5Cboxed%7B-%5Cdfrac%7Bx%2B7%7D%7By%7D%7D)
On the other hand, simplifying expression (c) gives something different.
![\dfrac{-x-7}{-y}=\dfrac{-(x+7)}{-(y)}=\dfrac{x+7}{y} \qquad\text{opposite the given expression}](https://tex.z-dn.net/?f=%5Cdfrac%7B-x-7%7D%7B-y%7D%3D%5Cdfrac%7B-%28x%2B7%29%7D%7B-%28y%29%7D%3D%5Cdfrac%7Bx%2B7%7D%7By%7D%20%5Cqquad%5Ctext%7Bopposite%20the%20given%20expression%7D)
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Another way you can write the expression is term-by-term with the terms in alpha-numeric sequence (so they're more easily compared).
Given: (-x-7)/y = (-x/y) +(-7/y)
(a) -(x+7)/y = (-x/y) +(-7/y)
(b) (x+7)/(-y) = (-x/y) +(-7/y)
(c) (-x-7)/(-y) = (x/y) +(7/y) . . . . not the same.
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Of course, you need to know the use of the distributive property and the rules of signs.
a(b+c) = ab +ac
-a/b = a/(-b) = -(a/b)
-a/(-b) = a/b
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<u>Summary</u>: The given expression matches (a) and (b).
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<em>Additional comments</em>
Sometimes, when I'm really stuck trying to see if two expressions are equal, I subtract one from the other. If the difference is zero, then I know they are the same. Looking at (b), we could compute ...
![\left(\dfrac{-x-7}{y}\right)-\left(\dfrac{x+7}{-y}\right)=\dfrac{-y(-x-7)-y(x+7)}{-y^2}\\\\=\dfrac{xy+7y-xy-7y}{-y^2}=\dfrac{0}{-y^2}=0](https://tex.z-dn.net/?f=%5Cleft%28%5Cdfrac%7B-x-7%7D%7By%7D%5Cright%29-%5Cleft%28%5Cdfrac%7Bx%2B7%7D%7B-y%7D%5Cright%29%3D%5Cdfrac%7B-y%28-x-7%29-y%28x%2B7%29%7D%7B-y%5E2%7D%5C%5C%5C%5C%3D%5Cdfrac%7Bxy%2B7y-xy-7y%7D%7B-y%5E2%7D%3D%5Cdfrac%7B0%7D%7B-y%5E2%7D%3D0)
Yet another way to check is to substitute numbers for the variables. It is a good idea to use (at least) one more set of numbers than there are variables, just to make sure you didn't accidentally find a solution where the expressions happen to be equal. We can use (x, y) = (1, 2), (2, 3), and (3, 5) for example.
The given expression evaluates to (-1-7)/2 = -4, (-2-7)/3 = -3, and (-3-7)/5 = -2.
(a) evaluates to -(1+7)/2 = -4, -(2+7)/3 = -3, -(3+7)/5 = -2, same as given
(b) evaluates to (1+7)/-2 = -4, (2+7)/-3 = -3, (3+7)/-5 = -2, same as given
(c) evaluates to (-1-7)/-2 = 4, different from given