In this problem, we need to find the correct system of linear equations that models the given situation about Jody's work hours and earnings.
Let's start by defining the variables:
- \( x \): the number of hours Jody babysat.
- \( y \): the number of hours Jody did yardwork.
According to the problem, Jody worked a total of 34 hours. This gives us the first equation:
[tex]\[ x + y = 34 \][/tex]
Next, we know that Jody earns \[tex]$10 per hour babysitting and \$[/tex]15 per hour doing yardwork. This week, she earned a total of \$410. Thus, we can write the second equation based on her earnings:
[tex]\[ 10x + 15y = 410 \][/tex]
Now, we need to identify the system of equations from the given options:
### Option A
[tex]\[ x + y = 34 \][/tex]
[tex]\[ 10x + 15y = 410 \][/tex]
### Option B
[tex]\[ x + y = 410 \][/tex]
[tex]\[ 10x + 15y = 34 \][/tex]
### Option C
[tex]\[ x + y = 34 \][/tex]
[tex]\[ 15x + 10y = 410 \][/tex]
### Option D
[tex]\[ x + y = 410 \][/tex]
[tex]\[ 15x + 10y = 34 \][/tex]
Comparing each option to our formulated system of equations:
- Option A has the system:
[tex]\[ x + y = 34 \][/tex]
[tex]\[ 10x + 15y = 410 \][/tex]
Which matches our equations.
- Option B has the system:
[tex]\[ x + y = 410 \][/tex]
[tex]\[ 10x + 15y = 34 \][/tex]
Which does not match our equations.
- Option C has the system:
[tex]\[ x + y = 34 \][/tex]
[tex]\[ 15x + 10y = 410 \][/tex]
Which has the coefficients of \( x \) and \( y \) swapped in the second equation. Therefore, it does not match.
- Option D has the system:
[tex]\[ x + y = 410 \][/tex]
[tex]\[ 15x + 10y = 34 \][/tex]
Which does not match our equations.
Based on our comparisons, Option A is the correct system of equations that models the situation.
Therefore, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
