Q.Passing through $(2, 2\sqrt{3})$ and inclined with the x-axis at an angle of $75^\circ$.

Slope Calculation — From Intuition to Precision

Imagine you're walking up a hill. Some hills are gentle — you barely notice the climb. Others are so steep you have to lean forward and use your hands. That "steepness" is what slope measures. In mathematics, slope tells us how fast a line rises or falls as we move from left to right.

The Intuition: Rise Over Run

Take any two points on a straight line. As you walk from the left point to the right point, two things happen:

• You move horizontally — that's the run.
• You move vertically — that's the rise (upwards) or fall (downwards).

Slope is simply the ratio:

Slope = (vertical change) ÷ (horizontal change)

If you climb 3 metres while walking 5 metres forward, the slope is $3/5 = 0.6$. If you descend 2 metres while walking 4 metres forward, the slope is $-2/4 = -0.5$ — negative because you're going downhill.

The Precise Definition

Given two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ on a non-vertical line, the slope $m$ is:

The numerator is the rise (change in $y$), the denominator is the run (change in $x$). The order matters: subtract the first point's coordinates from the second's, consistently.

What the Number Tells You

The larger the absolute value $|m|$, the steeper the line. A slope of $5$ is much steeper than a slope of $0.2$.

A Worked Example

Find the slope of the line through $(1, 2)$ and $(4, 8)$.

Step 1: Label the points. Let $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (4, 8)$.

Step 2: Compute the rise: $y_2 - y_1 = 8 - 2 = 6$.

Step 3: Compute the run: $x_2 - x_1 = 4 - 1 = 3$.

Step 4: Divide: $m = \frac{6}{3} = 2$.

The line rises 2 units vertically for every 1 unit it moves right. …