Gradient measures steepness: The gradient m is how much the line rises for each step across — rise ÷ run. Read two points off the line and divide the change in y by the change in x.
- one point on the line
- another point on the line
IB-style question — gradient from two points
Find the gradient of the line through A(1, 2) and B(4, 11).
Step by step
- Subtract the y-values (top) and the x-values (bottom) in the same order.
- Work it out.
Final answer
m = 3 (the line rises 3 for every 1 across).
What the sign tells you
- m > 0 → uphill (left to right).
- m < 0 → downhill.
- m = 0 → horizontal line y = c.
Watch out
- Subtract in the same order top and bottom.
- A vertical line x = a has no gradient (run = 0).
- rise/run, never run/rise.
Same line, three outfits: A straight line can be written three ways — pick whichever fits the question.
Gradient–intercept
- m = gradient
- c = y-intercept
- Best for graphing
Point–gradient
- Use a point + gradient
- Best for building a line
General form
- Tidy integer form
- Gradient = −a/b
IB-style question — switch between forms
Write y − 3 = 2(x − 1) in the form y = mx + c, then in the form ax + by + d = 0.
Step by step
- Expand the bracket.
- Make y the subject → gradient–intercept form.
- Move everything to one side → general form.
Final answer
y = 2x + 1, or equivalently 2x − y + 1 = 0.
Gradient straight from ax + by + d = 0: Rearrange to y = mx + c: the gradient is m = −a/b. (e.g. 3x + 2y − 6 = 0 → y = −1.5x + 3, so m = −1.5.) This exact step shows up in real exams.
Get feedback like a real examiner
Submit your answers and get instant feedback — what you did well, what's missing, and exactly what to write to score full marks.
Point + gradient, or two points: Given a gradient and a point, drop them straight into point–gradient form. Given two points, find the gradient first, then do the same.
IB-style question — a point and a gradient
Find the equation of the line through (2, 5) with gradient 3. Give your answer as y = mx + c.
Step by step
- Point–gradient form with (x₁, y₁) = (2, 5), m = 3.
- Expand and tidy.
Final answer
y = 3x − 1.
IB-style question — two points
Find the equation of the line through P(1, 2) and Q(3, 8).
Step by step
- Gradient first.
- Use point–gradient with either point, say (1, 2).
- Tidy.
Final answer
y = 3x − 1.
Either point works: With two points you can substitute either one into point–gradient form — you get the same line. Pick the one with smaller numbers.
Set the other coordinate to zero: y-intercept: set x = 0. x-intercept: set y = 0 and solve. (And in y = mx + c, the number c is the y-intercept — read it straight off.)
IB-style question — both intercepts
Find where the line y = 2x − 6 crosses each axis.
Step by step
- y-intercept: put x = 0.
- x-intercept: put y = 0 and solve.
Final answer
Crosses the y-axis at (0, −6) and the x-axis at (3, 0).
Don't swap the coordinates: The y-intercept is the point (0, c) and the x-intercept is (x, 0) — the zero goes in different slots. A common slip is writing (−6, 0) for the y-intercept.