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Slope Slope

Slope = how steep a line is

Slope tells you how much a line goes up or down as you move to the right.

Main formula
Pick two points: (x1, y1) and (x2, y2)
m = (y2 − y1) ÷ (x2 − x1)

Positive slope

x y 0 1 2 3 4 5 0 1 2 3 4 5 (1,1) (3,5) Positive slope Arrow check Move right → Arrow points up Slope is positive
Example (coordinates)
Points: \((1,1)\) and \((3,5)\)
\(m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{5 - 1}{3 - 1} = \dfrac{4}{2} = 2\)

Rise over run

Another way to remember slope: m = rise ÷ run.
Rise is the vertical change \((y_2 - y_1)\). Run is the horizontal change \((x_2 - x_1)\).

Note: “Rise” can go up or down. If it goes down, the rise is negative — it’s a fall. Glitch in math :)
x y 0 1 2 3 4 5 1 2 3 4 5 (1,1) (3,5) run = 2 rise = 4 Rise over run
Example (rise/run)
Points: \((1,1)\) and \((3,5)\)
\(\text{rise} = 5 - 1 = 4\), \(\text{run} = 3 - 1 = 2\)
\(m = \dfrac{\text{rise}}{\text{run}} = \dfrac{4}{2} = 2\)
One more
Points: \((0,2)\) and \((4,6)\)
\(\text{rise} = 6 - 2 = 4\), \(\text{run} = 4 - 0 = 4\)
\(m = \dfrac{4}{4} = 1\)

Negative slope

x y 0 1 2 3 4 5 0 1 2 3 4 5 (1,5) (3,1) Negative slope Arrow check Move right → Arrow points down Slope is negative
Example (coordinates)
Points: \((1,5)\) and \((3,1)\)
\(m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{1 - 5}{3 - 1} = \dfrac{-4}{2} = -2\)

Rise over run

Same idea, but the rise can be negative (down). Run is still “move right”.

x y 0 1 2 3 4 5 1 2 3 4 5 (1,5) (3,1) run = 2 rise = −4 Rise over run
Example (rise/run)
Points: \((1,5)\) and \((3,1)\)
\(\text{rise} = 1 - 5 = -4\), \(\text{run} = 3 - 1 = 2\)
\(m = \dfrac{-4}{2} = -2\)
One more
Points: \((0,4)\) and \((5,1)\)
\(\text{rise} = 1 - 4 = -3\), \(\text{run} = 5 - 0 = 5\)
\(m = \dfrac{-3}{5}\)

Quick check: if the line goes up as you move right → slope is positive.
If it goes down as you move right → slope is negative.

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