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Heather spent a total of 70 minutes working out at the gym. She spent 20 minutes longer running on the treadmill than she spent lifting weights.

When [tex]$t$[/tex] represents the time Heather spent running on the treadmill and [tex]$w$[/tex] represents the time she spent lifting weights, which system of equations represents this situation?

A. [tex]t - w = 20[/tex]
[tex]w = t + 70[/tex]

B. [tex]t - w = 20[/tex]
[tex]t = w + 70[/tex]

C. [tex]t + w = 70[/tex]
[tex]t = w + 20[/tex]

D. [tex]t + w = 70[/tex]
[tex]w = t + 20[/tex]

Sagot :

GinnyAnswer
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.
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