Lines, Gradients, and Real Life Maths

Year 10 Mathematics Understanding Linear Equations and Gradients Real-world Applications

What is a Linear Equation?

An equation that creates a straight line when graphed General form: y = mx + c m represents the gradient (slope) c represents the y-intercept

Understanding Gradient (Slope)

Gradient = rise ÷ run Measures how steep a line is Positive gradient: line goes up from left to right Negative gradient: line goes down from left to right Zero gradient: horizontal line

Calculate the Gradient

Given two points: (2, 3) and (6, 11) Use the formula: m = (y₂ - y₁) ÷ (x₂ - x₁) Work through the calculation step by step Answer: m = 2

Real Life Example: Wheelchair Ramps

{"left":"Building codes require wheelchair ramps to have a maximum gradient of 1:12\nThis means for every 12 units horizontal, the ramp can rise 1 unit","right":"Gradient = 1/12 ≈ 0.083\nSteeper ramps are unsafe and difficult to use"}

Think About It

A ski slope has a gradient of -0.5 What does this tell us about the slope? Is it going uphill or downhill? How steep is it compared to a gradient of -2?

Real Life Example: Phone Data Plans

Many phone plans charge a base fee plus cost per GB Example: $30 base + $5 per GB Linear equation: Cost = 5 × GB + 30 Gradient = 5 (cost increases by $5 per GB) Y-intercept = 30 (base cost when GB = 0)

Gradient in Nature and Engineering

Your Turn: Taxi Fare Problem

A taxi charges $4 base fare + $2.50 per km Write the linear equation for total cost What is the gradient and what does it represent? How much would a 8km trip cost?

Key Takeaways

Linear equations create straight lines with equation y = mx + c Gradient (m) shows the rate of change or steepness Y-intercept (c) shows the starting value Gradients appear everywhere: ramps, costs, slopes, growth rates Understanding gradients helps us analyze real-world relationships