5.5 Modelling with Straight Lines
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Learning Objectives
- Use linear equations to model real-world relationships
- Interpret the meaning of gradients and intercepts in context
- Construct models from data and assess their suitability
Key Concepts
What Is a Linear Model?
- A **linear model** is an equation of the form y = mx + c that approximates how one variable depends on another.
- Common in real-world contexts like economics, biology, physics, and business.
- Example: Cost = 5x + 20 (x = number of items, 5 = cost per item, 20 = fixed cost).
Interpreting Gradient and Intercept
- **Gradient (m)**: The rate of change or how much y increases/decreases when x increases by 1.
- Example: If m = 3, for each 1 unit increase in x, y increases by 3.
- **Y-Intercept (c)**: The value of y when x = 0 (often a starting value or fixed cost).
- Example: y = 3x + 7 → when x = 0, y = 7.
Constructing Linear Models from Data
- Given a table of values or scatter diagram, you can:
- Estimate a line of best fit (visually or using regression)
- Identify the gradient and intercept from patterns or formulas
- Form a linear equation y = mx + c to represent the trend
- Useful in predicting values or finding relationships.
Testing and Evaluating the Model
- Models are simplifications – they work **within a certain range**.
- Check how well the model fits data:
- Does the line pass near most points?
- Are there outliers or curves that the line doesn't capture?
- Always consider **limitations** and **assumptions** (e.g., constant rate).
Applications of Linear Modelling
- Estimating fuel consumption vs distance
- Predicting population growth over short time frames
- Modelling profit, cost, or temperature changes
- Used in statistics for correlation and regression analysis
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