8. \( \mathrm{d}=3.5 \mathrm{t} \)

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!