To find slope, you just need to measure how steep a line is: how much it goes up or down when you go to the right.

Key idea: “rise over run”

  • Slope is defined as
    slope=m=change in ychange in x=ΔyΔx\text{slope}=m=\dfrac{\text{change in }y}{\text{change in }x}=\dfrac{\Delta y}{\Delta x}slope=m=change in xchange in y​=ΔxΔy​.
  • “Rise” = change in vertical direction (y).
  • “Run” = change in horizontal direction (x).

Think of walking up a hill: if you go 2 steps up while going 1 step forward, the slope is 2/1=22/1=22/1=2.

1. From two points

If you know two points on a line, say (x1,y1)(x_1,y_1)(x1​,y1​) and (x2,y2)(x_2,y_2)(x2​,y2​), use:

m=y2−y1x2−x1m=\dfrac{y_2-y_1}{x_2-x_1}m=x2​−x1​y2​−y1​​

Steps:

  1. Label the points: first point (x1,y1)(x_1,y_1)(x1​,y1​), second point (x2,y2)(x_2,y_2)(x2​,y2​).
  1. Subtract the y’s: y2−y1y_2-y_1y2​−y1​ (this is your rise).
  1. Subtract the x’s: x2−x1x_2-x_1x2​−x1​ (this is your run).
  1. Divide rise by run and simplify.

Example: Points (5,8)(5,8)(5,8) and (1,4)(1,4)(1,4):

  • y2−y1=4−8=−4y_2-y_1=4-8=-4y2​−y1​=4−8=−4
  • x2−x1=1−5=−4x_2-x_1=1-5=-4x2​−x1​=1−5=−4
  • m=−4−4=1m=\dfrac{-4}{-4}=1m=−4−4​=1

So the slope of the line through those points is 1.

2. From a graph

When you see the line on a coordinate grid:

  1. Pick two clear points on the line (where it crosses grid intersections).
  1. Count how far up/down you go from the first point to the second (rise).
  2. Count how far right you go from the first point to the second (run).
  1. Then m=riserunm=\dfrac{\text{rise}}{\text{run}}m=runrise​.

Example picture in your head:

  • From point A to point B you go up 3 units and right 2 units.
  • Slope m=3/2m=3/2m=3/2.

If you go down instead of up, the rise is negative, so the slope is negative (the line slants down as you move right).

3. From an equation

Many linear equations are written in slope-intercept form:

y=mx+by=mx+by=mx+b

  • mmm is the slope.
  • bbb is the y‑intercept (where the line crosses the y‑axis).

Example:

  • Equation: y=7x+10y=7x+10y=7x+10
  • Coefficient of xxx is 7, so slope m=7m=7m=7.

If the equation is not in that form (for example 6x−2y=126x-2y=126x−2y=12):

  1. Solve for yyy to rewrite it as y=mx+by=mx+by=mx+b.
  1. The coefficient of xxx is the slope.

For 6x−2y=126x-2y=126x−2y=12:

  • Rearrange to get y=3x−6y=3x-6y=3x−6.
  • Slope is m=3m=3m=3.

4. Special slopes

  • Horizontal line : looks like y=cy=cy=c.
    • Rise =0=0=0, run ≠0\neq 0=0, so slope =0=0=0.
  • Vertical line : looks like x=cx=cx=c.
    • Run =0=0=0, division by zero is impossible, so slope is undefined.

These show up a lot in practice problems about “zero slope” or “undefined slope.”

5. Quick memory tricks

  • “Rise over run ” → always change in y over change in x, never the other way around.
  • Positive slope: line goes up as you move right.
  • Negative slope: line goes down as you move right.
  • The steeper the line, the bigger the absolute value of the slope (e.g., ∣5∣|5|∣5∣ is steeper than ∣1/2∣|1/2|∣1/2∣).

TL;DR:

  • With two points: m=y2−y1x2−x1m=\dfrac{y_2-y_1}{x_2-x_1}m=x2​−x1​y2​−y1​​.
  • From a graph: count rise and run, then do rise/run.
  • From an equation: rewrite as y=mx+by=mx+by=mx+b and read off the coefficient of xxx.

Bottom note: Information gathered from public forums or data available on the internet and portrayed here.