The Slope of a Line | Moosa Academy

Slope measures how steeply a line rises or falls. It is what makes two lines parallel — equal slopes — and it is calculated from nothing more than the coordinates of any two points on the line.

Concept The formula

m = \frac{y_2 - y_1}{x_2 - x_1}

Divide the difference in the $y$ -coordinates by the difference in the $x$ -coordinates. Choose any two points on the line and apply it.

Slope is the rise divided by the run — how far the line climbs for each step it takes sideways.

Note Keep the order consistent

Whichever point you call $(x_2,\ y_2)$ , use it first in both the numerator and the denominator. A common habit is to start from the point further right along the $x$ -axis and subtract the left-hand point from it.

Mixing the order — subtracting one way on top and the other way underneath — flips the sign and gives a slope that leans the wrong way.

Example A line through the origin

Find the slope of the line through $(0,\ 0)$ and $(1,\ 1)$ .

Difference in

y

1 - 0 = 1

Difference in

x

1 - 0 = 1

m = \dfrac{1}{1} = 1

⟹ the slope is 1

A slope of 1 means the line rises one unit vertically for every unit it moves horizontally.

Example Different points, same line

Using the points $(2,\ 2)$ and $(3,\ 3)$ on that same line:

Difference in

y

3 - 2 = 1

Difference in

x

3 - 2 = 1

m = \dfrac{1}{1} = 1

⟹ the slope is 1 again

The choice of points makes no difference. A straight line has a single slope everywhere along it — which is precisely what makes it straight.

Concept Slope and parallel lines

Two parallel lines must have the same slope. If one slope changes while the other stays fixed, the lines are no longer heading in the same direction and will eventually cross.

So checking whether two lines are parallel comes down to a calculation: work out both slopes and compare them.

Note Mistakes to avoid

Putting the

x

-difference on top — the rise goes over the run, not the other way round.

Subtracting in one order on top and the opposite order underneath.

Assuming different pairs of points give different slopes for the same line.

Losing a minus sign when a coordinate is negative.

Concluding that two lines are parallel without actually comparing their slopes.

Summary

Slope m = (y₂ − y₁) / (x₂ − x₁) — the rise divided by the run.
Keep the subtraction in the same order in the numerator and denominator.
Any two points on the line give the same slope.
A slope of 1 means one unit up for every unit across.
Parallel lines have equal slopes, so comparing slopes settles the question.