5.5PureStretch

Modelling with Straight Lines

A linear model uses $y = m x + c$ to describe a real relationship. The gradient becomes a rate (cost per item, speed) and the intercept a starting value (fixed cost, initial amount) — and interpreting them is the key exam skill.

25 min Video by Zeeshan Zamurred Straight Line Graphs

Edexcel Straight Lines — full chapter playlist (Zeeshan Zamurred)Watch the full walkthrough before the notes below.

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What you'll be able to do

Set up a linear model from a situation
Interpret the gradient in context
Interpret the intercept in context
Use the model to make predictions

Gradient as a rate

In a linear model, the gradient is the $rate of change$ — how much $y$ increases per unit of $x$ . For a cost model, that is the cost per extra item; for distance–time, it is speed.

1It is the gradient — cost per km.

Answer£2 per kilometre

Intercept as a starting value

The intercept $c$ is the value of $y$ when $x = 0$ — the $fixed$ or starting amount.

1It is the cost when

d = 0

— a fixed charge.

Answer£3 fixed charge

Tip — Gradient = “per” (rate); intercept = “fixed/start” (value when x = 0).

Making predictions

Substitute a value into the model to predict an output, but be wary of $extrapolating$ far beyond the data — a linear model may not hold there.

C = 2 (10) + 3

Answer£23

Formula recap

y = m x + c

Linear model.

m = rate of change (per unit)

Gradient meaning.

c = value when x = 0

Intercept meaning.

Common mistakes to avoid

Swapping the meanings of gradient and intercept.

Gradient = rate (per unit); intercept = fixed/starting value.

Extrapolating a linear model far beyond the data.

Predictions are reliable only within (or near) the data range.

Key takeaways

Linear models use y = mx + c.
Gradient = rate of change (per unit); intercept = starting/fixed value.
Substitute to predict; beware extrapolating beyond the data.

Test yourself

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