To solve this problem, we need to translate the given point [tex]\((5, -7)\)[/tex] by the specified translations: [tex]\(x \rightarrow x + 4\)[/tex] and [tex]\(y \rightarrow y - 5\)[/tex].
Let's go through the steps in detail:
1. Identify the initial coordinates:
- The initial [tex]\(x\)[/tex]-coordinate is [tex]\(5\)[/tex].
- The initial [tex]\(y\)[/tex]-coordinate is [tex]\(-7\)[/tex].
2. Apply the translation to the [tex]\(x\)[/tex]-coordinate:
- According to the problem, the [tex]\(x\)[/tex]-coordinate should be translated by adding [tex]\(4\)[/tex].
- So, the new [tex]\(x\)[/tex]-coordinate will be:
[tex]\[
x_{\text{new}} = 5 + 4 = 9
\][/tex]
3. Apply the translation to the [tex]\(y\)[/tex]-coordinate:
- According to the problem, the [tex]\(y\)[/tex]-coordinate should be translated by subtracting [tex]\(5\)[/tex].
- So, the new [tex]\(y\)[/tex]-coordinate will be:
[tex]\[
y_{\text{new}} = -7 - 5 = -12
\][/tex]
Therefore, the new coordinates of the point after the translation will be [tex]\((9, -12)\)[/tex].
