Gradient Between Two Coordinates GCSE: Rise Over Run Formula

What does gradient measure?

Gradient measures the steepness of a straight line — how much y changes for every 1-unit increase in x. It is also called the slope.

Gradient = rise ÷ run = (vertical change) ÷ (horizontal change)

The formula with coordinates:

m = (y₂ − y₁) ÷ (x₂ − x₁)

It does not matter which point you label (x₁, y₁) and which you label (x₂, y₂), as long as you use the same point for both the y-part and the x-part of the calculation.

How do you apply the formula?

Method at a glance:

1. Label the two points. Write (x₁, y₁) for the first and (x₂, y₂) for the second.
2. Calculate the change in y. Subtract: y₂ − y₁.
3. Calculate the change in x. Subtract: x₂ − x₁ (using the same order).
4. Divide. m = (y₂ − y₁) ÷ (x₂ − x₁).
5. Simplify the fraction if possible. Write the gradient as a fraction or decimal.

Worked examples

Example 1: Find the gradient of the line joining A(2, 5) and B(6, 13).

• Change in y: 13 − 5 = 8
• Change in x: 6 − 2 = 4
• m = 8 ÷ 4 = 2

The line rises by 2 units for every 1 unit to the right. ✓

Example 2: Find the gradient of the line joining P(−1, 7) and Q(3, −1).

• Change in y: −1 − 7 = −8
• Change in x: 3 − (−1) = 4
• m = −8 ÷ 4 = −2

The negative value confirms the line falls from left to right. ✓

Example 3: Find the gradient of the line joining C(0, 4) and D(5, 4).

• Change in y: 4 − 4 = 0
• Change in x: 5 − 0 = 5
• m = 0 ÷ 5 = 0

Zero gradient means the line is horizontal. ✓

Example 4: Find the gradient of the line joining E(3, 1) and F(3, 8).

• Change in y: 8 − 1 = 7
• Change in x: 3 − 3 = 0
• m = 7 ÷ 0 — undefined

Zero in the denominator means the line is vertical and has no defined gradient. ✓

What do different gradient values tell you?

Why does it not matter which point you call (x₁, y₁)?

If you swap the labels, both the numerator and denominator change sign:

Using A(2, 5) and B(6, 13) the other way: m = (5 − 13) ÷ (2 − 6) = −8 ÷ −4 = 2. Same answer. ✓

The crucial rule: both subtractions must use the same order — if you put the second point's y-value first in the numerator, you must also put the second point's x-value first in the denominator.

How does the formula connect to y = mx + c?

Once you have the gradient m from two points, you can find the equation of the line. Substitute one of the known points into y = mx + c and solve for c. This gives the full equation of the straight line through both points.

Example (continuing Example 1): m = 2; point A(2, 5). 5 = 2(2) + c → 5 = 4 + c → c = 1. Equation: y = 2x + 1 ✓ Check with B(6, 13): 2(6) + 1 = 13 ✓

Frequently asked questions

Can the gradient be a fraction?

Yes — fractions are common. For example, the gradient between (1, 3) and (4, 5) is (5 − 3) ÷ (4 − 1) = 2/3. Leave the answer as a fraction unless the question asks for a decimal.

What if the two points are the same?

If both coordinates are identical, the "line" is just a point — gradient is undefined in that case. This will not happen in a well-posed exam question.

How is gradient different from "rate of change"?

They are the same thing in different contexts. On a distance–time graph, gradient equals speed. On a cost–quantity graph, gradient equals cost per item. "Gradient" is the mathematical term; "rate of change" is used when the axes have real-world units.

If the gradient is negative, does the formula still give the right answer?

Yes — as long as you keep the sign throughout. Take care when subtracting a negative: x₂ − x₁ = 3 − (−1) = 3 + 1 = 4, not 3 − 1 = 2. Missing the double-negative step is the most common error in gradient calculations.

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