Ch 2 · 3.5–3.7 Equations of Lines, Parallel and Perpendicular Lines · O Level Mathematics 4024

Topic 2 of 2 in Coordinate Geometry — O Level Mathematics 4024 — 56 lessons.

Introduction

Every straight line has an equation that tells you its steepness and where it crosses the y-axis. Once you can find that equation, you can find lines parallel to it (same direction) or perpendicular to it (at right angles), which is how roads, ramps and boundaries are designed. (Syllabus sections 3.5–3.7.)

Key ideas — y = mx + c

m is the gradient and c the y-intercept. Rearrange other forms to read them: 5x + 4y = 8 ⇒ y = -tfrac54x + 2, gradient -tfrac54, intercept 2. Vertical lines are x = k; horizontal lines y = k.

🧑‍🏫 Teacher's explanation

Think of a straight line as a path on a map. The gradient, m, tells you how steep the path is—whether you are climbing up a hill or sliding down into a valley. The y-intercept, c, is simply the starting height where the path crosses the vertical axis.

📖 Definitions

🔍 Worked example — Easy

Find the gradient and y-intercept of the line with equation 3x - 2y = 6.

🔍 Worked example — Standard

Find the equation of the line parallel to y = 4x - 1 that passes through the point (2, 5).

🔍 Worked example — Exam-style

Find the equation of the perpendicular bisector of the line segment joining A(2, 4) and B(6, 10).

⚠️ Common errors

🧠 Quick check 1

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🎯 Exam question

Key ideas — Finding an equation

From a gradient m and a point (x_1, y_1): substitute into y = mx + c to find c (or use y - y_1 = m(x - x_1)). From two points: find m first. From a graph: read the intercept and measure the gradient. Give the final equation fully simplified, in the form asked.

🧑‍🏫 Teacher's explanation

Think of a straight line as a set of rules that tells every point (x, y) exactly where to stand. The 'gradient' m is the rule for how steep the line is—how much it rises for every step it moves forward. The 'y-intercept' c is simply the starting height where the line crosses the vertical axis.

📖 Definitions

🔍 Worked example — Easy

Find the equation of the line with gradient 3 that passes through the point (2, 5).

🔍 Worked example — Standard

Find the equation of the line perpendicular to y = 2/3x + 4 that passes through (4, 1).

🔍 Worked example — Exam-style

Find the equation of the perpendicular bisector of the line segment joining A(-2, 4) and B(4, 8).

⚠️ Common errors

🧠 Quick check 1

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🎯 Exam question

Key ideas — Parallel lines

Parallel lines have equal gradients. The line parallel to y = 4x - 1 through (1, -3): -3 = 4(1) + c ⇒ c = -7, so y = 4x - 7.

🧑‍🏫 Teacher's explanation

Imagine you are walking up a staircase. If two staircases are built with the exact same steepness, they are parallel; no matter how far you walk along them, the distance between them stays exactly the same. In coordinate geometry, 'steepness' is what we call the gradient, denoted by m.

📖 Definitions

🔍 Worked example — Easy

Find the gradient of a line that is parallel to the line with equation y = -3x + 5.

🔍 Worked example — Standard

Find the equation of the line parallel to 2y = 6x + 4 that passes through the point (2, 5).

🔍 Worked example — Exam-style

Find the equation of the line parallel to 3x - 4y = 12 that passes through the point (-2, 1). Give your answer in the form ax + by = c.

⚠️ Common errors

🧠 Quick check 1

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🎯 Exam question

Key ideas — Perpendicular lines

Perpendicular gradients multiply to -1:

🧑‍🏫 Teacher's explanation

Imagine two roads crossing each other at a perfect 90^° angle, like a crossroad in a city grid. If one road is climbing steeply, the road crossing it must be descending to maintain that perfect right angle. This relationship is captured by the gradients of the two lines.

📖 Definitions

🔍 Worked example — Easy

Find the gradient of a line that is perpendicular to the line with equation y = 4x - 7.

🔍 Worked example — Exam-style

Find the equation of the perpendicular bisector of the line segment joining A(2, 3) and B(6, 11).

⚠️ Common errors

🧠 Quick check 1

🧠 Quick check 2

🎯 Exam question

Key ideas — Perpendicular bisector

The perpendicular bisector of AB passes through the midpoint of AB with gradient -dfrac1m_AB.

🧑‍🏫 Teacher's explanation

Imagine you are standing exactly halfway between two friends who are standing apart. If you want to walk in a direction that is perfectly 'square' or at a right angle to the line connecting them, you are walking along their perpendicular bisector. It is the unique path that stays exactly equidistant from both people.

Ch 2 · 3.5–3.7 Equations of Lines, Parallel and Perpendicular Lines · O Level Mathematics 4024

All chapters in Coordinate Geometry