Find the equation of the straight line through (−1, 8) and (3, −4).
- Gradient: m = (−4 − 8)/(3 − (−1)) = −12/4 = −3.
- Put (−1, 8) into y = −3x + c: 8 = −3 × (−1) + c = 3 + c, so c = 5.
- Check with (3, −4): −3 × 3 + 5 = −4.
Answer: y = −3x + 5
Coordinates and straight-line graphs
Maths · Algebra. 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
A straight-line graph is fixed by two things: how steep it is and where it crosses the y-axis. In y = mx + c, m is the gradient and c is the y-intercept. The gradient is how far the line goes up for each 1 unit across; a negative gradient slopes down from left to right.
You can read m and c only when the equation is in the form y = ..., so rearrange first.
Parallel lines have the same gradient. Perpendicular lines meet at 90° and their gradients multiply to −1. For two points you can also find the midpoint (average the coordinates) and the distance between them (Pythagoras).
Rules to remember
- Gradient = change in y ÷ change in x (m = (y2 − y1)/(x2 − x1)).
- Midpoint = ((x1 + x2)/2, (y1 + y2)/2).
- Length of a line segment = √((x2 − x1)2 + (y2 − y1)2).
- Parallel lines: same m. Perpendicular lines: m1 × m2 = −1, so the new gradient is −1/m (turn it upside down and change the sign).
- Line with gradient m through (x1, y1): y − y1 = m(x − x1), or put the point into y = mx + c to find c.
- A line crosses the y-axis where x = 0 and the x-axis where y = 0.
- In a real context, m is the rate (for example cost per km) and c is the starting value (for example a fixed charge).
Common mistake
Reading the gradient before rearranging. For 2y = 6x + 8 the gradient is 3, not 6, because the equation must first be divided by 2 to give y = 3x + 4.
Worked example
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