What happens if you add 5 to a number and then subtract 5 from the result?
The same as the starting position. See solution.
Practice makes perfect
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.
First Translation:& (x-5, y + 7)
Second Translation:& (x+5, y - 7)
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-coordinate of this point becomes x_1 - 5 and the y-coordinate becomes y_1 +7.
(x_1, y_1) → (x_1 - 5, 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).
(x_1-5, y_1+7) → (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.
(x_1 -5 +5, y_1 + 7-7) = (x_1, 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.
Extra
Graphing the Translation
Let's consider an example figure and graph both translations!
