Apply m = (y₂-y₁)/(x₂-x₁) to find slope.

Find Slope of Line Passing Through (2,3) and (5,9)

Question

Find the slope of the line passing through the points (2, 3) and (5, 9).

Solution — Step by Step

Label the points clearly: $(x_1, y_1) = (2, 3)$ and $(x_2, y_2) = (5, 9)$ .

Labelling before substituting is a small habit that prevents sign errors — students who skip this step almost always flip a value somewhere.

The slope formula is:

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

Substituting our values:

m = \frac{9 - 3}{5 - 2} = \frac{6}{3} = 2

The slope $m = 2$ means the line rises 2 units vertically for every 1 unit it moves horizontally. Since $m > 0$ , the line slopes upward from left to right.

Answer: $m = 2$

Why This Works

The slope formula measures rate of change — how fast $y$ changes relative to $x$ as we move along the line. The numerator $(y_2 - y_1)$ is the vertical change (rise), and the denominator $(x_2 - x_1)$ is the horizontal change (run). Together, rise over run gives us the steepness and direction of the line.

What makes this formula powerful is that it gives the same result regardless of which two points on the line you pick. Any two points on a straight line will always yield the same slope — that’s precisely what makes it a straight line. Try picking a third point on this line (say $(8, 15)$ ) and compute the slope with $(2, 3)$ — you’ll get $m = 2$ again.

This concept is foundational for everything that follows in straight lines: equation of a line, angle between two lines, condition for parallel and perpendicular lines. Nail the slope calculation first, and the rest follows naturally.

Alternative Method — Using the Angle Approach

If you’re given the angle $\theta$ that the line makes with the positive $x$ -axis, you can find slope using $m = \tan\theta$ . This is more useful when the geometric picture is given rather than coordinates.

For verification here: we found $m = 2$ , so $\theta = \tan^{-1}(2) \approx 63.4°$ . The line cuts the $x$ -axis at a steep angle — consistent with a slope of 2. You won’t need this for this specific question, but in JEE problems, they often give $\theta$ directly and expect you to connect it to the slope.

When the line passes through the origin, $\theta$ and $m$ together define the line completely — no intercept needed. This shortcut saves 30 seconds in time-bound JEE Main questions.

Common Mistake

Subtracting in the wrong order. A very common error is writing $m = \frac{y_1 - y_2}{x_2 - x_1}$ — mixing up which point comes first in numerator vs denominator. This gives $m = \frac{3-9}{5-2} = \frac{-6}{3} = -2$ , which is completely wrong in sign.

The rule: whatever order you subtract in the numerator, use the same order in the denominator. If you start with $y_2 - y_1$ on top, you must write $x_2 - x_1$ on the bottom. Either order works — just be consistent.

Question

Solution — Step by Step

Why This Works

Alternative Method — Using the Angle Approach

Common Mistake

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