WHY define it as a ratio and not just "rise"?
Because "rise" alone is meaningless without saying how far you ran. Climbing 3 metres over 1 metre horizontally is steep; climbing 3 metres over 100 metres is nearly flat. The ratio normalises rise per unit run, so it measures true steepness.
Take any two distinct points on a line: A=(x1,y1) and B=(x2,y2).
Step 1 — measure the horizontal change (run).
Moving from A to B, x goes from x1 to x2, so
Δx=x2−x1.Why this step? Run is just the horizontal displacement; displacement is always (final − initial).
Step 2 — measure the vertical change (rise).
Similarly y goes from y1 to y2, so
Δy=y2−y1.Why this step? Rise is the vertical displacement, again (final − initial), and we must subtract in the same order as we did for x.
Step 3 — take the ratio.m=x2−x1y2−y1,x2=x1.Why this step? Slope is defined as rise per run, so we divide Step 2 by Step 1.
WHY is the answer independent of which two points you pick?
On a straight line, the triangles formed by different point-pairs are similar (same angle to the horizontal). Similar right triangles have equal ratios of corresponding sides, so Δy/Δx is the same everywhere. This is exactly what makes "the slope of a line" a well-defined single number.
Link to angle. If a line makes angle θ with the positive x-axis, then in the right triangle rise =Δy is "opposite" and run =Δx is "adjacent", so
m=ΔxΔy=adjacentopposite=tanθ.Why? Tangent is opposite/adjacent by definition — slope is the tangent of the inclination angle.
Define slope in words. → ratio of rise to run, Δy/Δx.
Which coordinate difference goes on top? → the y's (rise).
Slope of a horizontal line? → 0. Vertical line? → undefined.
Slope =tan of what? → the angle θ the line makes with the positive x-axis.
Why is slope the same for any two points on a line? → similar triangles ⇒ equal side ratios.
Recall Feynman: explain to a 12-year-old
Imagine walking up a ramp. Slope is a "cost tag": how many steps up do you get for each step forward? If the tag says 2, then every step forward makes you climb 2 up — steep! If it says 0, you never go up — flat floor. If the tag says a minus number, you're going downhill. And a wall going straight up? You can't "step forward" at all, so the cost tag makes no sense — that's why we call a vertical line's slope "undefined." To find the tag, pick two spots, see how much higher the second is (rise), see how much farther across it is (run), and divide.
Dekho, slope ka matlab bas ek simple sawaal hai: "line kitni steep hai aur kis direction me jaa rahi hai?" Jab hum right ki taraf chalte hain (x badhata hai), to line kitna upar ya neeche jaati hai (y ka change) — usi ki ratio ko slope kehte hain. Formula hai m=x2−x1y2−y1, yaani rise upon run, y top pe aur x bottom pe.
Sabse important cheez: subtraction ka order same rakhna. Agar y me pehle B liya to x me bhi pehle B lo, warna sign ulta ho jaayega. Positive slope matlab line uphill (right jaate hue upar), negative matlab downhill, zero matlab flat horizontal line, aur agar x2=x1 ho gaya to run zero, division by zero — matlab vertical line, jiska slope undefined hota hai (zero nahi, yaad rakhna!).
Ek aur mast baat: slope kisi bhi do points se nikaalo, answer same aayega, kyunki similar triangles ban-te hain — isliye ek line ka ek hi slope hota hai. Aur m=tanθ, jahan θ line ka angle hai x-axis se. Yeh cheez aage bahut kaam aati hai — straight line equation y=mx+c, parallel/perpendicular lines, aur calculus me derivative bhi to slope hi hai. Isliye is concept ko rise/run ke feel ke saath achhe se pakad lo.
