Ever wondered why the line on a graph looks just like the path a car would take on a straight road? That simple picture hides a lot of useful maths, especially when you need to ace your ISC Coordinate Geometry chapter.
A straight line is just a set of points that line up perfectly, like beads on a string. Its equation is a short code that tells you exactly where the line is on the x‑y grid.
What does the equation of a straight line tell us?
In coordinate geometry, a line is described by an equation – a mathematical sentence that connects any point (x, y) on the line. The most common way to write it is y = mx + c. Here m is the slope (also called gradient), which measures how steep the line is, and c is the y‑intercept, the point where the line crosses the y‑axis.
Understanding slope (gradient)
The slope tells you how much y changes when x moves by one unit. Think of water flowing through a pipe: the steeper the pipe, the faster the water rises or falls. Mathematically, slope = (change in y) ÷ (change in x) = (y₂‑y₁)/(x₂‑x₁). A positive slope means the line rises, a negative slope means it falls, and a slope of zero means the line is flat.
Worked example: finding the slope
Find the slope of the line passing through (2, 3) and (5, 11).
Using the formula:
m = (11‑3)/(5‑2) = 8/3 ≈ 2.67.
So for every 1 unit you move right, the line jumps up about 2.67 units.
Common forms of a straight‑line equation
Depending on the information you have, you might use one of four handy versions.
| Form | When to use | Equation |
|---|---|---|
| Slope‑intercept | you know slope (m) and y‑intercept (c) | y = mx + c |
| Point‑slope | you know slope (m) and a point (x₁, y₁) | y - y₁ = m(x - x₁) |
| Two‑point | you know two points on the line | (y - y₁)/(x - x₁) = (y₂ - y₁)/(x₂ - x₁) |
| General (ax + by + c = 0) | any line, especially when coefficients are integers | ax + by + c = 0 |
Example: converting to slope‑intercept form
Given 2x - 3y + 6 = 0, rewrite it as y = mx + c.
First isolate y:
‑3y = -2x - 6 → y = (2/3)x + 2.
So the slope m = 2/3 and the y‑intercept c = 2.
Intercepts: where the line meets the axes
The x‑intercept is the point where y = 0; the y‑intercept is where x = 0. Plugging zero into the appropriate variable gives you the other coordinate.
For y = 4x - 8:
- y‑intercept: set x = 0 → y = -8 → (0, -8).
- x‑intercept: set y = 0 → 0 = 4x - 8 → x = 2 → (2, 0).
Parallel and perpendicular lines
Two lines are parallel if they never meet; that happens when their slopes are equal (m₁ = m₂). They are perpendicular if they cross at a right angle; then the product of their slopes is –1 (m₁·m₂ = –1). This rule is a quick way to check if a line you draw will be at a right angle to another.
Worked example: equation of a line perpendicular to 3x - 4y + 5 = 0 and passing through (1, 2)
First write the given line in slope‑intercept form:
‑4y = -3x - 5 → y = (3/4)x + 5/4, so its slope m₁ = 3/4.
Perpendicular slope m₂ = -1 / (3/4) = -4/3.
Use point‑slope with point (1, 2):
y - 2 = -4/3 (x - 1) → multiply by 3: 3y - 6 = -4x + 4 → 4x + 3y - 10 = 0.
Quick reference summary
- Slope (m) = rise/run = (y₂‑y₁)/(x₂‑x₁).
- y‑intercept (c) = value of y when x = 0.
- Slope‑intercept form: y = mx + c.
- Point‑slope form: y - y₁ = m(x - x₁).
- Two‑point form: (y - y₁)/(x - x₁) = (y₂ - y₁)/(x₂ - x₁).
- General form: ax + by + c = 0.
- Parallel lines → same slope.
- Perpendicular lines → product of slopes = –1.
📝 Likely Exam Questions
- Find the equation of the line passing through (‑2, 5) and having a slope of 3.
Answer: Using point‑slope: y‑5 = 3(x+2) → y = 3x + 11. - Write the given line 5x + 2y – 7 = 0 in slope‑intercept form and state its slope and y‑intercept.
Answer: 2y = -5x + 7 → y = -(5/2)x + 7/2. Slope = -5/2, y‑intercept = 7/2. - Two lines are given: L₁: y = 2x + 1 and L₂: 4x - ky + 8 = 0. Find k so that L₁ ⟂ L₂ (perpendicular).
Answer: Convert L₂: -ky = -4x - 8 → y = (4/k)x + 8/k. Slope of L₂ = 4/k. For perpendicular: (2)*(4/k) = -1 → 8/k = -1 → k = -8. - Determine the x‑ and y‑intercepts of the line 3x - 6y + 12 = 0.
Answer: y‑intercept (x=0): -6y + 12 = 0 → y = 2 → (0, 2). x‑intercept (y=0): 3x + 12 = 0 → x = -4 → (-4, 0).