# Straight-Line Graphs

> CIE IGCSE Additional Mathematics · CIE IGCSE Add Maths (0606)
> Source: https://www.owlsprep.com/study/cie-0606-u7-overview/
> Weight: 8-10% of total assessment, appearing in both Paper 1 (non-calculator, structured written) and Paper 2 (calculator, structured written)

This unit covers core coordinate geometry concepts for straight-line graphs, from basic distance and gradient calculations to converting non-linear relationships into straight-line form using linear law, a high-weight exam skill for CIE IGCSE Additional Mathematics.

**Prerequisites:** Basic algebraic rearrangement; Familiarity with Cartesian coordinate systems

## Learning objectives

- Calculate gradients, midpoints, and lengths of line segments between pairs of Cartesian coordinates
- Use and convert between all standard forms of straight-line equations, including parallel and perpendicular line rules
- Transform non-linear relationships into straight-line form using linear law to identify unknown constants
- Solve exam-style coordinate geometry problems involving straight-line graphs and linear law applications

## 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](https://www.owlsprep.com/study/cie-0606-u7-straight-lines-gradients-midpoint-and/) — Learn to calculate gradients, midpoints, line segment lengths, and use all forms of straight-line equations, including parallel and perpendicular line rules.
- [Linear Law — Straight-Line Form](https://www.owlsprep.com/study/cie-0606-u7-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.

## Common pitfalls

- **Wrong:** Confusing perpendicular and parallel gradient rules
  - Why it fails: Parallel lines have equal gradients, while perpendicular lines have gradients whose product is -1, a frequent mix-up in exam questions.
  - Correct: 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:** Forgetting to rearrange non-linear terms correctly when applying linear law
  - Why it fails: Linear law requires isolating constant terms to match $Y = mX + c$ form, with Y and X as modified derived variables.
  - Correct: Clearly label your modified Y and X variables before rearranging to avoid mixing up gradient and intercept values.
- **Wrong:** Using unmodified raw coordinates when plotting linear law graphs
  - Why it fails: Linear law graph axes use transformed Y and X variables, not the original x and y values from the non-linear equation.
  - Correct: Always plot transformed terms on the axes, then use the resulting line's gradient and intercept to solve for original constants.

## 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 | $L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ | Straight Lines, Gradients, Midpoint and Length |
| Midpoint of line segment | $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$ | Straight Lines, Gradients, Midpoint and Length |
| Slope-intercept line form | $y = mx + c$ 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 $Y = mX + c$ | 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.

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