To solve this problem, we need to apply the given translations to the initial position of the point step by step. Let's start by identifying the initial position and the translations:
1. Initial Position:
[tex]\((0, -2)\)[/tex]
2. First Translation Vector:
[tex]\(\langle 3, 5 \rangle\)[/tex]
3. Second Translation Vector:
[tex]\(\langle 0, -1 \rangle\)[/tex]
### Step-by-Step Solution:
#### Step 1: Apply the First Translation
We start by applying the first translation vector [tex]\(\langle 3, 5 \rangle\)[/tex] to the initial position [tex]\((0, -2)\)[/tex].
Adding the components of the first translation vector to the initial position:
[tex]\[
\begin{aligned}
x' &= 0 + 3 = 3 \\
y' &= -2 + 5 = 3 \\
\end{aligned}
\][/tex]
So, after the first translation, the point's new position is:
[tex]\[
(3, 3)
\][/tex]
#### Step 2: Apply the Second Translation
Next, we apply the second translation vector [tex]\(\langle 0, -1 \rangle\)[/tex] to the position obtained after the first translation, which is [tex]\((3, 3)\)[/tex].
Adding the components of the second translation vector to the position obtained after the first translation:
[tex]\[
\begin{aligned}
x'' &= 3 + 0 = 3 \\
y'' &= 3 - 1 = 2 \\
\end{aligned}
\][/tex]
So, after the second translation, the point's final position is:
[tex]\[
(3, 2)
\][/tex]
### Final Answer:
The ordered pair for the point's final position after undergoing the translations [tex]\(T_{\langle 3,5 \rangle}\)[/tex] and [tex]\(T_{\langle 0,-1 \rangle}\)[/tex] starting from [tex]\((0, -2)\)[/tex] is:
[tex]\[
(3, 2)
\][/tex]
