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G5.14 The Gradient of a Line

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01 — Explanation

What does "gradient" actually measure?

Gradient measures steepness as "rise over run" — the vertical change divided by the horizontal change between two points on a line:

m = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} = \frac{y_2-y_1}{x_2-x_1}

A gradient of 3 means: for every 1 unit moved right, the line rises 3 units. In context, gradient is often a real rate of change — speed on a distance–time graph, hourly wage on an earnings graph, or exchange rate on a currency-conversion graph.

02 — Explanation

What do the four types of gradient look like?

Type	Appearance	Condition
Positive	uphill, left to right	$m>0$
Negative	downhill, left to right	$m<0$
Zero	perfectly flat	$m=0$ (equation $y=c$ )
Undefined	perfectly vertical	run $=0$ , so division by zero (equation $x=k$ )

Exam Tip

A horizontal line is flat because every point shares the same $y$ -value, making the rise zero — a vertical line is undefined because every point shares the same $x$ -value, making the run (the denominator) zero.

03 — Explanation

How do I calculate the gradient from two coordinates?

Use $m=\dfrac{y_2-y_1}{x_2-x_1}$ , bracketing negative values. Example: between $(7,-1)$ and $(4,5)$ : $m=\dfrac{5-(-1)}{4-7}=\dfrac{6}{-3}=-2$ .

Exam Tip

It doesn't matter which point you call "1" and which "2" — but you must subtract $y$ -values and $x$ -values in the same order. Calculating $\dfrac{y_2-y_1}{x_1-x_2}$ (mismatched order) is a very common error that flips the sign.

04 — Explanation

How do I read the gradient straight off an equation?

If the equation is in the form $y=mx+c$ , the gradient is just the coefficient of $x$ . If it isn't in that form, rearrange first.

Example: Find the gradient of $5x+2y=7$ .

2y=-5x+7 \;\Rightarrow\; y=-\frac52x+\frac72

Gradient $=-\dfrac52$ .

05 — Explanation

How do gradients relate between parallel and perpendicular lines?

Parallel lines share the same gradient: $m_1=m_2$ .
Perpendicular lines have gradients that multiply to $-1$ : $m_1\times m_2=-1$ . To find a perpendicular gradient, flip the fraction and change its sign (the "negative reciprocal"). If $m=\tfrac23$ , the perpendicular gradient is $-\tfrac32$ .

Worked Example

Find the equation of the line perpendicular to $y=\tfrac12x+1$ that passes through $(10,6)$ .

Gradient of the given line: $m_1=\tfrac12$ .
Perpendicular gradient: $m_2=-\dfrac{1}{1/2}=-2$ .
New line: $y=-2x+c$ . Substitute $(10,6)$ : $6=-2(10)+c \Rightarrow c=26$ .

Equation: $y=-2x+26$ .

Exam Tip

For a curve, the "gradient at a point" means the gradient of the tangent line drawn at that exact point — read two clear points off the tangent and apply the same rise-over-run formula.

06 — Explanation

How do I draw a line accurately given a point and a gradient?

Plot the given point first. Then write the gradient as a fraction $\dfrac{\text{rise}}{\text{run}}$ (e.g. $m=-2$ becomes $\dfrac{-2}{1}$ ), move that many units horizontally and vertically from your point to mark a second point, and draw a straight line through both. A positive rise means moving up; a negative rise means moving down.

Example: draw a line of gradient $\dfrac23$ through $(0,-1)$ . From $(0,-1)$ , move 3 right and 2 up to reach $(3,1)$ , then rule a straight line through both points.

Exam Tip

When reading a gradient from a graph, always pick two points that land exactly on grid-line intersections — estimating from points that don't sit on a clean intersection is a common source of small but costly inaccuracies.