To solve this problem, we need to set up a system of equations based on the information given. Let's carefully interpret the problem statement:
1. Heather spent a total of 70 minutes working out at the gym.
2. She spent 20 minutes longer running on the treadmill than she spent lifting weights.
We are given two unknowns:
- [tex]\( t \)[/tex]: the time Heather spent running on the treadmill
- [tex]\( w \)[/tex]: the time Heather spent lifting weights
First, let's translate the total time spent at the gym into an equation:
[tex]\[ t + w = 70 \][/tex]
This equation represents the total time Heather spent working out.
Next, we address the information that she spent 20 minutes longer on the treadmill than lifting weights. This gives us another equation:
[tex]\[ t = w + 20 \][/tex]
This states that the time spent on the treadmill [tex]\( t \)[/tex] is 20 minutes more than the time spent lifting weights [tex]\( w \)[/tex].
Therefore, the system of equations that correctly represents the situation is:
[tex]\[ t + w = 70 \][/tex]
[tex]\[ t = w + 20 \][/tex]
Looking at the given options:
- Option A states:
[tex]\[ t + w = 70 \][/tex]
[tex]\[ w = t + 20 \][/tex]
- Option B states:
[tex]\[ t - w = 20 \][/tex]
[tex]\[ w = t + 70 \][/tex]
- Option C states:
[tex]\[ t - w = 20 \][/tex]
[tex]\[ t = w + 70 \][/tex]
- Option D states:
[tex]\[ t + w = 70 \][/tex]
[tex]\[ t = w + 20 \][/tex]
Comparing the statements with our derived system, we find that Option D matches perfectly.
Thus, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
Option D. [tex]\( t + w = 70 \)[/tex] and [tex]\( t = w + 20 \)[/tex]
