Lesson Guide
Explanation
If you know the gradient and one point, use the point gradient formula: y minus y one equals m times, x minus x one, where x one, y one is the point. For gradient three through two, five: y minus five equals three times, x minus two. Expand: y minus five equals three x minus six. Add five to both sides: y equals three x minus one. Check it: when x is two, y is six minus one, which is five.
Your Turn
Gradient −2 through (1, 4). Check: y − 4 = −2(x − 1), so y = −2x + 6.
Watch Out
You subtract the point's coordinates. Through (−3, 2) gives y − 2 = m(x + 3), not m(x − 3).
Explanation
Another way is to substitute the point straight into y equals m x plus c. With gradient negative one half, the line is y equals negative one half x, plus c. It passes through four, negative three, so put x equals four and y equals negative three: negative three equals negative two plus c. So c is negative one, and the line is y equals negative one half x, minus one.
Your Turn
Gradient 4 through (−1, 1). Check: 1 = −4 + c, so c = 5 and y = 4x + 5.
Watch Out
The point's x goes in for x and its y goes in for y. Swapping them gives the wrong c.
Explanation
With two points, find the gradient first. From negative one, negative four to three, four, the rise is eight and the run is four, so the gradient is two. Now use either point to find c. With three, four: four equals two times three, plus c, so c is negative two. The line is y equals two x minus two. Always check with the other point: two times negative one, minus two, is negative four, which is correct.
Your Turn
Through (0, 3) and (2, −1). Check: m = −2 and c = 3, so y = −2x + 3.
Watch Out
Find m before c. Then check with the point you did not use: it catches most slips.
Explanation
To find the equation of a line from its graph, read the y-intercept first. This line crosses the y-axis at one, so c is one. Then find two points where the line passes exactly through a corner of the grid, such as zero, one and four, negative two. From one to the other, the run is four and the rise is negative three, so the gradient is negative three quarters. The equation is y equals negative three quarters x, plus one.
Your Turn
A line crosses the y-axis at −2 and passes through (3, 0). Check: m = \(\dfrac{2}{3}\), so y = \(\dfrac{2}{3}\)x − 2.
Watch Out
Count the run from left to right. A line that falls to the right has a negative gradient, whatever your counting gave you.
Explanation
A point is on a line exactly when its coordinates make the equation true. Does five, seven lie on y equals two x minus three? Two times five, minus three, is seven, and y is seven, so yes. Does negative two, negative six? Two times negative two, minus three, is negative seven, not negative six, so no. Close is not enough. It works for any form: for three x minus four y equals twelve, the point eight, three gives twenty four minus twelve, which is twelve, so it is on the line. Quiz and exercises now.
Your Turn
Is (−1, 9) on y = −4x + 5? Check: 4 + 5 = 9, so yes.
Watch Out
Substitute both coordinates and compare. Working out the right-hand side is only half the test.
📌 Key Points — this lesson in a nutshell
- A line with gradient m through (x1, y1) is y − y1 = m(x − x1). Expand it and make y the subject.
- Or put m into y = mx + c, substitute the point, and solve for c.
- From two points: find the gradient first, then c from either point. Check with the other point.
- From a graph: read c on the y-axis, then the gradient between two grid-corner points.
- A point lies on a line only if both its coordinates make the equation true.
Step 1: Introduction
Step 1 of 5
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