Linear Relationships – When Change Is Constant

A linear relationship exists between two quantities when one changes by a constant amount every time the other increases by one unit. If you plotted the pairs of values, they would line up in a perfectly straight line – hence the name. This constant, unchanging rate of change is the single defining feature that separates linear relationships from every other kind.

Linear relationships are everywhere: a taxi fare that rises by a fixed amount per kilometre, a candle that burns down by the same height every hour, a savings account earning the same fixed deposit every month. Recognising this pattern is the foundation for everything else in this section – tables, graphs, equations, and gradients are all just different ways of describing the same underlying idea.

Recognising a Linear Relationship

Real-Life Application

• Taxi fares: a starting fee plus a fixed rate per kilometre is a linear relationship.
• Savings: depositing the same fixed amount every week creates a linear relationship over time.
• Fuel consumption: a car using fuel at a constant rate has a linear relationship between distance and fuel remaining.

Key Takeaways

Practice: Linear Relationships