Unit Overview

Straight-Line Graphs

CIE IGCSE Additional Mathematics· 5 min read 📊 8-10% of total assessment, appearing in both Paper 1 (non-calculator, structured written) and Paper 2 (calculator, structured written)

1. Unit at a glance

You will start with foundational straight-line calculations: gradient between two points, midpoint of a segment, length of a line segment, and different forms of straight-line equations (y = mx + c, ax + by + c = 0, two-point form), including rules for parallel and perpendicular lines.

You will then move to linear law, a practical skill that lets you transform non-linear relationships into plotable straight lines to identify unknown constants, a commonly tested structured question topic in exams.

Work through the following subtopics in order to build your mastery:

Straight Lines, Gradients, Midpoint and Length

Learn to calculate gradients, midpoints, line segment lengths, and use all forms of straight-line equations, including parallel and perpendicular line rules.

★★⏱ 12 min

Linear Law — Straight-Line Form

Master converting non-linear equations to straight-line form, plotting transformed graphs, and solving for unknown constants using linear law.

★★★⏱ 15 min

2. Common Pitfalls

Wrong move:

Confusing perpendicular and parallel gradient rules

Why:

Parallel lines have equal gradients, while perpendicular lines have gradients whose product is -1, a frequent mix-up in exam questions.

Correct move:

Always verify: if $m_{1} = m_{2}$ , lines are parallel; if $m_{1} \times m_{2} = - 1$ , lines are perpendicular (excluding horizontal/vertical edge cases).

Wrong move:

Forgetting to rearrange non-linear terms correctly when applying linear law

Why:

Linear law requires isolating constant terms to match $Y = m X + c$ form, with Y and X as modified derived variables.

Correct move:

Clearly label your modified Y and X variables before rearranging to avoid mixing up gradient and intercept values.

Wrong move:

Using unmodified raw coordinates when plotting linear law graphs

Why:

Linear law graph axes use transformed Y and X variables, not the original x and y values from the non-linear equation.

Correct move:

Always plot transformed terms on the axes, then use the resulting line's gradient and intercept to solve for original constants.

3. Quick Reference Cheatsheet

Concept	Formula/Rule	Subtopic
Gradient between $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$	$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$	Straight Lines, Gradients, Midpoint and Length
Length of line segment	$L = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$	Straight Lines, Gradients, Midpoint and Length
Midpoint of line segment	$M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$	Straight Lines, Gradients, Midpoint and Length
Slope-intercept line form	$y = m x + c$ where $m$ = gradient, $c$ = y-intercept	Straight Lines, Gradients, Midpoint and Length
Perpendicular lines rule	$m_{1} \times m_{2} = - 1$	Straight Lines, Gradients, Midpoint and Length
Linear law general form	Rewrite non-linear $f (y) = A f (x) + B$ as $Y = m X + c$	Linear Law — Straight-Line Form

What's Next

Start with the first subtopic to build foundational straight-line calculation skills, which you will apply immediately in the linear law subtopic. Mastery of this unit is required for upcoming coordinate geometry units including circles, as well as kinematics problems in later sections of the syllabus.