Select the correct answer.

To solve the problem, we need to establish the two equations based on the information given.

1. The total number of hours worked by Jody is 34. This can be expressed as:
\[
x + y = 34
\]
where \( x \) represents the hours she babysat and \( y \) represents the hours she did yardwork.

2. The total earnings from babysitting at $10 per hour and yardwork at $15 per hour sums to $410. This can be expressed as:
\[
10x + 15y = 410
\]

Putting these two equations together forms the system:
\[
\begin{align*}
1. & \quad x + y = 34 \\
2. & \quad 10x + 15y = 410
\end{align*}
\]

Now, looking at the answer choices provided:

- A. \( x + y = 34 \) and \( 10x + 15y = 410 \) (This matches our derived equations.)
- B. \( x + y = 410 \) and \( 10x + 15y = 34 \) (Incorrect)
- C. \( x + y = 34 \) and \( 15x + 10y = 410 \) (Incorrect)
- D. \( x + y = 410 \) and \( 15x + 10y = 34 \) (Incorrect)

Thus, the correct answer is:
**A.**
\( x + y = 34 \)
\( 10x + 15y = 410 \)