To solve the problem, let's write the system of equations that models Jody's work situation.
1. Expressing the total hours worked:
Jody worked a total of 34 hours this week. If [tex]\( x \)[/tex] represents the number of hours she babysat and [tex]\( y \)[/tex] represents the number of hours she did yardwork, then the sum of these hours should equal 34. Hence, our first equation is:
[tex]\[
x + y = 34
\][/tex]
2. Expressing the total earnings:
Jody earns [tex]\(\$ 10\)[/tex] per hour for babysitting and [tex]\(\$ 15\)[/tex] per hour for yardwork. Given that her total earnings for the week are [tex]\(\$ 410\)[/tex], we can write this information as:
[tex]\[
10x + 15y = 410
\][/tex]
Therefore, the system of equations that models this situation is:
[tex]\[
\begin{cases}
x + y = 34 \\
10x + 15y = 410
\end{cases}
\][/tex]
Comparing this with the given options:
A. [tex]\( x + y = 34 \)[/tex]
[tex]\[
10x + 15y = 410
\][/tex]
B. [tex]\( x + y = 410 \)[/tex]
[tex]\[
10x + 15y = 34
\][/tex]
C. [tex]\( x + y = 34 \)[/tex]
[tex]\[
15x + 10y = 410
\][/tex]
D.
[tex]\[
\begin{array}{l}
x + y = 410 \\
15x + 10y = 34
\end{array}
\][/tex]
The correct system of equations is option A:
[tex]\[
\begin{cases}
x + y = 34 \\
10x + 15y = 410
\end{cases}
\][/tex]
So, the answer is [tex]\( \boxed{A} \)[/tex].
