Exercise 9 Page 458

We are told that a figure is translated by $(x-5,y+7)$ and then translated again, this time by $(x+5,y-7).$ We are asked to determine the final position of the figure without graphing and explain our reasoning. Let's consider what happens to the figure after the first translation. Imagine the point $(x_1,y_1)$ is a part of the figure. Since the translation is $(x-5,y+7),$ the $x\text{-}$coordinate of this point becomes $x_1-5$ and the $y\text{-}$coordinate becomes $y_1+7.$ The second translation is applied to the image of the first translation. In the case of our point, this means that it is not applied to the point $(x_1,y_1),$ but to the point $(x_1-5,y_1+7).$ The translation is $(x+5,y-7),$ so our point becomes $(x_1-5+5,y_1+7-7).$ Note that, since addition and subtraction are inverse operations, the result of subtracting $5$ from $x_1$ and then adding $5$ is $x_1.$ Similarly, the result of adding $7$ to $y_1$ and then subtracting $7$ is $y_1.$ After applying the two translations to the point $(x_1,y_1)$ we get the point $(x_1,y_1).$ The same is true for any other point that we apply these two translations to. Therefore, the final position of our figure is the same as the starting position of the figure.