Let's analyze the problem step-by-step to determine the correct system of equations.
1. Understanding the Variables:
- Let [tex]\( t \)[/tex] be the time Heather spent running on the treadmill.
- Let [tex]\( w \)[/tex] be the time she spent lifting weights.
2. Translating the Problem into Mathematical Expressions:
- Heather spent 20 minutes longer running on the treadmill than lifting weights.
- This can be expressed as [tex]\( t = w + 20 \)[/tex].
- The total time Heather spent working out is 70 minutes.
- This can be expressed as [tex]\( t + w = 70 \)[/tex].
3. Identifying the Correct System of Equations:
- Based on the problem, the two key equations are:
1. [tex]\( t = w + 20 \)[/tex] (which states that Heather spent 20 minutes longer running than lifting weights).
2. [tex]\( t + w = 70 \)[/tex] (which states the total workout time).
4. Matching this with the Given Choices:
- Let's compare this with each provided option:
A. [tex]\( t - w = 20 \)[/tex] and [tex]\( w = t + 70 \)[/tex]
- The second equation is incorrect because it suggests that [tex]\( w \)[/tex] is 70 minutes more than [tex]\( t \)[/tex], which contradicts the total time of 70 minutes.
B. [tex]\( t - w = 20 \)[/tex] and [tex]\( t = w + 70 \)[/tex]
- The second equation is incorrect because it does not align with the relationship stated in the problem and implies Heather was running far longer than the total available time.
C. [tex]\( t + w = 70 \)[/tex] and [tex]\( t = w + 20 \)[/tex]
- Both equations accurately reflect the information given.
- The first equation confirms the total workout time.
- The second equation correctly states that Heather spent 20 minutes longer running than lifting weights.
D. [tex]\( t + w = 70 \)[/tex] and [tex]\( w = t + 20 \)[/tex]
- The second equation incorrectly reverses the relationship between [tex]\( t \)[/tex] and [tex]\( w \)[/tex].
Given the analysis, the correct system of equations that represents Heather's workout time is:
[tex]\[
C. \quad t + w = 70 \quad \text{and} \quad t = w + 20
\][/tex]
Thus, the correct answer is option C.
