The Gradient of a Straight Line

Work out how steep a straight line is from two points or from its equation, and tell uphill, downhill and flat lines apart.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel IGCSE Maths 4MA1, IGCSE Additional Maths 0606, Edexcel Further Pure Maths 4PM1 · 31 real past-paper questions below

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What Gradient Means

The gradient tells you how steep a line is. It is how far the line goes up for every 1 you go across.

Formula

m = \frac{up}{across} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

$m$ is the gradient. $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ are any two points on the line.

Line	Gradient
Goes up to the right	positive
Goes down to the right	negative
Flat	zero
Straight up (upright)	no gradient: across is 0, and you cannot divide by 0

Key idea

A bigger number means a steeper line. A gradient of 4 is steeper than a gradient of 1.

Key idea

In many countries, including the UK and USA, a wheelchair ramp may be no steeper than 1 : 12: 1 unit up for every 12 across. That is a gradient of $\frac{1}{12}$ , about $0.08$ , gentle enough to push up safely.

Gradient From Two Points

The 2-mark question: work out the gradient of the line through two points. Take the same point first on the top and the bottom.

Worked example

1
$A (- 1, 5), B (3, - 7)$ — The two points. Call them A and B so you keep the same order all the way through.
2
$m = \frac{- 7 - 5}{3 - ( - 1 )}$ — Write the fraction first: y of B minus y of A on top, x of B minus x of A underneath.
3
$- 7 - 5 = - 12$ — Work out the top. From 5 down to −7 is a drop of 12, so up is −12.
4
$3 - (- 1) = 3 + 1$ — Work out the bottom. Taking away a negative is adding: $- (- 1) = + 1$ .
5
$3 + 1 = 4$ — So across is 4.
6
$m = \frac{- 12}{4} = - 3$ — Divide up by across. It is negative because the line goes down to the right.

Key idea

Show the fraction line, like $\frac{- 7 - 5}{3 - ( - 1 )}$ . It earns 1 of the 2 marks on its own, even if the sum after it slips.

Common mistake

Doing $5 - (- 7)$ on top but $3 - (- 1)$ underneath. That mixes the order and gives 3. Start with the same point both times.

Gradient From a Line on a Grid

Many questions draw the line on a grid and ask for its gradient. Read two points from the axes, then do up ÷ across.

Worked example

1
$(- 1, 6) and (3, - 2)$ — Pick two points where the line passes exactly through a grid corner, far apart. Exact points give an exact answer.
2
$y-axis: 0, 2, 4, 6$ — Read the numbers on each axis. Here the y-axis goes up in 2s, so one square up is 2, not 1.
3
$- 2 - 6 = - 8$ — Up: y of the second point minus y of the first. It drops 8, so up is −8.
4
$3 - (- 1) = 4$ — Across: x of the second minus x of the first. Minus a negative is plus, so 3 + 1 = 4.
5
$m = \frac{- 8}{4} = - 2$ — Divide up by across.
6
$down to the right \Rightarrow m < 0$ — Check the sign with your eyes: the line falls to the right, so a negative gradient is right.

Common mistake

Counting squares: 4 down and 4 across gives −1. The y-axis goes up in 2s, so 4 squares is 8. Always use the numbers on the axes.

Gradient From an Equation

The 1-mark question gives $y = 4 - 2 x$ . When y is on its own, the gradient is the number in front of $x$ , with its sign: here $- 2$ . In $y = \frac{x}{4} + 1$ , $\frac{x}{4}$ means $\frac{1}{4} x$ , so the gradient is $\frac{1}{4}$ .

The 2-mark question gives an equation like $3 x + 2 y = 8$ . Get y on its own first.

Worked example

1
$3 x + 2 y = 8$ — The question. x and y are on the same side, so the 3 is not the gradient yet.
2
$2 y = - 3 x + 8$ — Take 3x from both sides, so only the y term is left on the left.
3
$y = - \frac{3}{2} x + 4$ — Divide every term by 2, including the 8, so y is on its own.
4
$m = - \frac{3}{2} = - 1.5$ — Now it is $y = m x + c$ , so the gradient is the number in front of x, minus sign included.

If y has a minus in front, divide by the negative number. Watch the signs.

Worked example

1
$2 x - 5 y = 10$ — The question. This time the y term is −5y.
2
$- 5 y = - 2 x + 10$ — Take 2x from both sides.
3
$y = \frac{2}{5} x - 2$ — Divide every term by −5. Negative ÷ negative is positive, so −2x becomes $+ \frac{2}{5} x$ , and 10 ÷ (−5) = −2.
4
$m = \frac{2}{5}$ — The number in front of x. Leave it as a fraction or write 0.4.

Common mistake

Answering 3. The 3 is not the gradient until y is on its own.

Find a Missing Coordinate

Some questions give the gradient and hide one coordinate. Put everything into the gradient formula, then solve for the letter.

Worked example

1
$P (2, k), Q (5, 4), m = - 3$ — The question: find k. Use Q first on the top and the bottom, as always.
2
$\frac{4 - k}{5 - 2} = - 3$ — Up ÷ across, set equal to the gradient you were given.
3
$\frac{4 - k}{3} = - 3$ — Work out the bottom: 5 − 2 = 3.
4
$4 - k = - 9$ — Multiply both sides by 3 to clear the fraction: −3 × 3 = −9.
5
$- k = - 13$ — Take 4 from both sides: −9 − 4 = −13.
6
$k = 13$ — Multiply both sides by −1 to make k positive.
7
$\frac{4 - 13}{3} = \frac{- 9}{3} = - 3$ — Check: put k = 13 back in. The gradient is −3, as given.

Key idea

Extended papers may use letters for both points. For gradient 2 through $(a, 5)$ and $(b, 13)$ : $\frac{13 - 5}{b - a} = 2$ , so $8 = 2 (b - a)$ , so $a = b - 4$ . Same moves.

Which Method? Read the Question

Look at what you are given.

You see	You do
Two points	up ÷ across
y on its own, like $y = 2 - 3 x$	number in front of x
x and y together, like $5 x + 2 y = 7$	get y on its own first
A line drawn on a grid	two exact points from the axes, then up ÷ across
The gradient, with one coordinate missing	put it all in up ÷ across and solve
The equation of a line drawn on a grid	Finding the Equation From a Graph
Same direction as another line	Parallel Lines and Their Equations
At right angles to a line	Perpendicular Lines

Key idea

Check the sign with a sketch: a line going down to the right must have a negative gradient.

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