8. $$ \mathrm{d}=3.5 \mathrm{t} $$

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Certainly! Let's analyze and understand the equation you've provided:

$$ \mathrm{d} = 3.5 \, \mathrm{t} $$

### Understanding the Equation

This equation represents a **linear relationship** between two variables:

- **$$\mathrm{d}$$**: Typically denotes **distance**.
- **$$\mathrm{t}$$**: Typically denotes **time**.
- **3.5**: This is the **rate** at which the distance changes with respect to time. In many contexts, this could represent **speed**.

### Possible Context

Assuming **$$\mathrm{d}$$** is distance and **$$\mathrm{t}$$** is time, the equation can be interpreted as:

$$ \text{Distance} = \text{Speed} \times \text{Time} $$

Here, the **speed** is **3.5 units per time unit** (e.g., kilometers per hour, meters per second, etc., depending on the context).

### Solving for One Variable in Terms of the Other

1. **If you want to find the distance ($$\mathrm{d}$$) traveled in a given time ($$\mathrm{t}$$):**

$$ \mathrm{d} = 3.5 \, \mathrm{t} $$

- **Example**: If $$ \mathrm{t} = 2 $$ hours,
     $$ \mathrm{d} = 3.5 \times 2 = 7 \text{ units} $$

2. **If you want to find the time ($$\mathrm{t}$$) it takes to travel a certain distance ($$\mathrm{d}$$):**

To solve for $$ \mathrm{t} $$, divide both sides of the equation by 3.5:

$$ \mathrm{t} = \frac{\mathrm{d}}{3.5} $$

- **Example**: If $$ \mathrm{d} = 10 $$ kilometers,
     $$ \mathrm{t} = \frac{10}{3.5} \approx 2.857 \text{ hours} $$

### Graphical Representation

If you graph this equation with:

- **Horizontal Axis (x-axis)**: Time ($$\mathrm{t}$$)
- **Vertical Axis (y-axis)**: Distance ($$\mathrm{d}$$)

The graph will be a straight line passing through the origin (0,0) with a slope of **3.5**. This slope represents the constant speed.

### Applications

Such linear equations are fundamental in various fields, including:

- **Physics**: Calculating distance traveled over time at constant speed.
- **Economics**: Modeling costs that increase linearly with production levels.
- **Engineering**: Predicting resource utilization over time.

### Summary

- **Equation**: $$ \mathrm{d} = 3.5 \, \mathrm{t} $$
- **Interpretation**: Distance is directly proportional to time with a proportionality constant (speed) of 3.5 units per time unit.
- **Solving for Distance**: Multiply the speed by time.
- **Solving for Time**: Divide the distance by the speed.

If you have a specific context or additional details related to this equation, feel free to share them, and I can provide a more tailored explanation!
The equation $$ \mathrm{d} = 3.5 \, \mathrm{t} $$ shows that distance ($$\mathrm{d}$$) is 3.5 times time ($$\mathrm{t}$$). This means the speed is 3.5 units per time unit.

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