Sure, let's find the image of the point [tex]\( P(-5, 7) \)[/tex] under the translation [tex]\( T(x, y) = (x+5, y-4) \)[/tex].
1. Identify the initial coordinates:
- [tex]\( x = -5 \)[/tex]
- [tex]\( y = 7 \)[/tex]
2. Apply the translation transformation to the [tex]\( x \)[/tex]-coordinate:
- The translation rule for the [tex]\( x \)[/tex]-coordinate is [tex]\( x' = x + 5 \)[/tex].
- Substitute [tex]\( x = -5 \)[/tex]:
[tex]\[
x' = -5 + 5 = 0
\][/tex]
3. Apply the translation transformation to the [tex]\( y \)[/tex]-coordinate:
- The translation rule for the [tex]\( y \)[/tex]-coordinate is [tex]\( y' = y - 4 \)[/tex].
- Substitute [tex]\( y = 7 \)[/tex]:
[tex]\[
y' = 7 - 4 = 3
\][/tex]
So, after applying the translation transformation, the new coordinates [tex]\( P' \)[/tex] are:
[tex]\[
P' = (0, 3)
\][/tex]
Therefore, the number that belongs in the green box is:
[tex]\[
\boxed{3}
\][/tex]
