Find the equation of the line through (−1, 6) and (3, −6).
- Gradient: m = (−6 − 6)/(3 − (−1)) = −12/4 = −3.
- Use (−1, 6) in y = −3x + c: 6 = −3 × (−1) + c = 3 + c, so c = 3.
- Check with (3, −6): −3 × 3 + 3 = −6.
Answer: y = −3x + 3
Equations of linear graphs
Maths · Coordinate geometry. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
Every straight line that is not vertical can be written as y = mx + c. Here m is the gradient and c is the y-intercept, the value of y where the line crosses the y-axis.
To find the equation of a line you need two facts: its gradient and one point on it. If you are given two points, find the gradient from them first. Then put the gradient and the coordinates of one point into y = mx + c to find c.
If an equation is in another form, such as 4x − 2y = 8, rearrange it to make y the subject before you read off m and c.
Rules to remember
- y = mx + c: m is the gradient, c is the y-intercept
- Gradient from two points: m = (y2 − y1)/(x2 − x1)
- To find c, substitute the gradient and one known point into y = mx + c
- Another form: y − y1 = m(x − x1) for a line with gradient m through (x1, y1)
- Horizontal line through (a, b): y = b; vertical line through (a, b): x = a
- Divide every term when making y the subject: 3y = 2x + 12 becomes y = (2/3)x + 4
Common mistake
Reading m and c before y is alone. In 3y = 2x + 12 the y-intercept is 4, not 12, because every term must be divided by 3.
Worked example
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