To complete the equation and sentences, we need to determine the relationship between the total exercise time $T$ and the distance Judy runs $d$. Here's how:

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Step 1: Identify the relationship

The total exercise time $T$ includes:

1. A constant stretching time (denoted as $S$).
2. A running time that depends on her pace (minutes per mile).

Step 2: Determine Judy's pace (rate of running)

The table shows that the additional time Judy spends running as the distance increases is consistent. For example:

• Between 2 and 4 miles: $51 - 33 = 18$ minutes for 2 miles → Pace: $\frac{18}{2} = 9$ minutes per mile.
• Between 4 and 7 miles: $78 - 51 = 27$ minutes for 3 miles → Pace: $\frac{27}{3} = 9$ minutes per mile.

Thus, her running pace is 9 minutes per mile.

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Step 3: Determine the stretching time ($S$)

From the table, subtract the running time for any distance to find the constant stretching time:

• At $d = 2$: Total time $T = 33$, Running time $2 \times 9 = 18$, so $S = 33 - 18 = 15$.
• At $d = 4$: Total time $T = 51$, Running time $4 \times 9 = 36$, so $S = 51 - 36 = 15$.

Thus, the stretching time is 15 minutes.

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Step 4: Write the equation

The total exercise time $T$ in minutes is the sum of the stretching time $S$ and the running time ($9d$, where $d$ is the distance in miles):

$T = 9d + 15$

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Step 5: Complete the sentences

1. Judy spends 9 minutes per mile running.
2. She spends 15 minutes stretching, no matter how far she runs.

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Final Answer

The equation is:

$T = 9d + 15$

Would you like a detailed explanation of how to solve similar problems or additional examples? 😊

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Related Questions

1. How would the equation change if Judy’s stretching time increased to 20 minutes?
2. What is Judy’s total exercise time if she runs 10 miles?
3. How far did Judy run if her total exercise time was 69 minutes?
4. What does the slope of the equation $T = 9d + 15$ represent in this context?
5. How would the graph of $T = 9d + 15$ look, and what does the y-intercept represent?

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Tip

When analyzing linear relationships in word problems, identify fixed (constant) values and rates of change to build the equation systematically.