Q.For what value of "$k$" the points $(k,7)$, $(-4,5)$ and $(1,-5)$ are collinear.

What does "slope" even mean?

Imagine you're walking up a hill. Some hills are gentle — you barely feel the climb. Others are steep — you're out of breath in seconds. That "steepness" is exactly what slope measures in mathematics. It tells you how fast a line rises (or falls) as you move from left to right.

A flat road has slope zero. A ramp that goes up quickly has a large slope. A ramp that goes down has a negative slope. That's the core idea: slope = steepness + direction.

From intuition to numbers

Suppose you're tracking the height of a plant over time. On day 1 it's 3 cm tall; on day 4 it's 9 cm tall. How fast is it growing?

The plant grew from 3 cm to 9 cm — that's a rise of $6$ cm. It took $3$ days — that's a run of $3$ days. The growth rate is:

That number $2$ is the slope. Every day, the height increases by 2 cm. If the slope were negative, the plant would be shrinking. If it were zero, the plant wouldn't be growing at all.

The precise formula

Given any two points on a line — say $(x_1, y_1)$ and $(x_2, y_2)$ — the slope $m$ is:

The numerator $y_2 - y_1$ is the vertical change (rise). The denominator $x_2 - x_1$ is the horizontal change (run). You're simply dividing how much the line goes up or down by how much it goes sideways.

The Greek letter $\Delta$ (delta) means "change in". So $\Delta y$ is "change in y", and $\Delta x$ is "change in x".

A worked example

Find the slope of the line through $(2, 5)$ and $(7, 20)$.

Label the points: let $(x_1, y_1) = (2, 5)$ and $(x_2, y_2) = (7, 20)$. Then:

The slope is $3$. That means for every 1 unit you move right, the line goes up 3 units. Steep upward climb.

What the sign tells you

• Positive slope ($m > 0$): the line rises as you go right. Like walking uphill.
• Negative slope ($m < 0$): the line falls as you go right. Like walking downhill.
• Zero slope ($m = 0$): the line is perfectly horizontal. No rise at all.
• Undefined slope: the line is vertical. You'd be dividing by zero ($\Delta x = 0$), which is impossible. A vertical line has no defined slope — it's infinitely steep.

Why it matters

The slope formula is the foundation of coordinate geometry. It lets you:

• Write the equation of a line ($y = mx + c$)
• Tell if two lines are parallel (same slope) or perpendicular (slopes multiply to $-1$)
• Find rates of change in physics, economics, and biology
• Understand derivatives in calculus — slope of a curve at a point

Once you see slope as "rise over run", you've unlocked the language of change.

The slope of a line through $(x_1, y_1)$ and $(x_2, y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$.