Equations of Straight Lines
Algebra · Equations of Straight Lines
Syllabus tag: KCSE | Mathematics | Form 2 | Topic 4 Equations of Straight Lines
Lesson objectives
By the end of this topic, you should be able to:
- Determine the gradient of a straight line through known points.
- Determine the equation of a straight line using the gradient and one point.
- Express a linear equation in the forms y = mx + c and ax + by + c = 0.
- Determine the gradients of parallel and perpendicular lines.
Equations of Straight Lines
Every straight line has an equation. It carries two facts: how steep the line is, and where it crosses the y-axis.
a) Gradient
The gradient measures steepness. It is the change in y divided by the change in x.
People remember it as rise over run.
Take the points in the same order top and bottom. Reversing one but not the other flips the sign.
A positive gradient rises left to right. A negative gradient falls.
A horizontal line has gradient 0. A vertical line has an undefined gradient, since the run is zero and division by zero is undefined.
b) The equation y = mx + c
In y = mx + c, the value m is the gradient. The value c is the y-intercept, where the line crosses the y-axis.
Reading a line's equation tells you both facts immediately.
For y = 3x − 4, the gradient is 3 and the line crosses the y-axis at −4.
c) Finding the equation
Given the gradient and one point, use y − y₁ = m(x − x₁).
Substitute, expand, and rearrange into the form the question asks for.
Given two points, find the gradient first, then use either point in the same formula. Both points give the same final equation.
d) The form ax + by + c = 0
The same line can be written with everything on one side.
From y = 2x + 1, subtract y to get 2x − y + 1 = 0.
Clear any fractions by multiplying through, and it is usual to keep the x coefficient positive.
Both forms describe the same line. Give whichever the question asks for.
e) Parallel lines
Parallel lines have equal gradients.
So y = 3x + 1 and y = 3x − 5 are parallel. They never meet, because they rise at the same rate.
For a line parallel to a given one through a stated point, keep the gradient. Then use the point-gradient form.
f) Perpendicular lines
The product of perpendicular gradients is −1.
So if one gradient is 2, the perpendicular gradient is −½.
Put another way, invert the gradient and change its sign.
A line of gradient 1 and one of gradient −1 meet at right angles, as the figure shows.
The exception is a horizontal and a vertical line. They are perpendicular, but the product rule fails because a vertical gradient is undefined.
g) Where this is used
Any linear relationship: cost against quantity, distance against time, temperature conversion. The gradient is always a rate, and the intercept always a fixed starting value.
Words to know
- Gradient -- the change in y divided by the change in x; the steepness of a line.
- y-intercept -- the value of y where a line crosses the y-axis.
- Parallel lines -- lines with equal gradients that never meet.
- Perpendicular lines -- lines meeting at right angles, with gradients multiplying to −1.
- General form -- the equation written as ax + by + c = 0.
:::checkpoint Check yourself
- Find the gradient of the line through (2, 5) and (6, 13).
- Find the equation of the line with gradient 3 passing through (1, 4).
- What is the gradient of a line perpendicular to y = 4x − 7?
- Why is the gradient of a vertical line undefined? :::
Bridge to practice
The exercises begin with gradient and its sign, move through the two standard forms and finding equations, and finish with parallel and perpendicular lines and applications. Before reading a gradient from any equation, rearrange it into y = mx + c first.