# Slope and Linear Equations — Rise Over Run
> **Animation coming** — full Sora 2 prompt below. Video will replace this placeholder.
## Plain English first
Slope is steepness. It answers: for every step you take to the right, how many steps do you go up (or down)?
If slope m = 2, every 1 step right means 2 steps up — a steep climb. If m = 0.5, every 1 step right means half a step up — a gentle slope. If m = −1, every step right means 1 step down — a descent. If m = 0, perfectly flat.
A linear equation y = mx + b just says: start at height b when x = 0, then change by m for every unit you move right. That's the whole thing.
---
## Standard math notation
```
Slope formula (given two points):
m = (y₂ − y₁) / (x₂ − x₁) = rise / run
Slope-intercept form:
y = mx + b
m = slope, b = y-intercept (height where line crosses the y-axis)
Point-slope form (given slope m and one point (x₁, y₁)):
y − y₁ = m(x − x₁)
Standard form:
Ax + By = C
Horizontal line: y = b (slope = 0, no rise)
Vertical line: x = a (slope = undefined, infinite rise per zero run)
Parallel lines: same slope m, different b
Perpendicular lines: slopes are negative reciprocals (m₁ × m₂ = −1)
```
---
## Verbose Python with descriptive names
```python
def compute_slope_from_two_points(
x_coordinate_of_first_point,
y_coordinate_of_first_point,
x_coordinate_of_second_point,
y_coordinate_of_second_point
):
"""
Slope = how much y changes per unit of x change.
Rise = vertical change. Run = horizontal change. Slope = rise / run.
"""
vertical_rise = y_coordinate_of_second_point - y_coordinate_of_first_point
horizontal_run = x_coordinate_of_second_point - x_coordinate_of_first_point
if horizontal_run == 0:
return None # vertical line — slope is undefined
slope = vertical_rise / horizontal_run
return slope
def compute_y_from_slope_intercept(
slope_of_line,
y_intercept,
x_value
):
"""
Given slope m and y-intercept b, find y at any x.
y = mx + b: start at b, add m for every unit of x.
"""
y_value = slope_of_line * x_value + y_intercept
return y_value
def compute_y_intercept_from_point_and_slope(
x_of_known_point,
y_of_known_point,
slope_of_line
):
"""
Given a point on the line and the slope, find where the line hits y-axis.
From y = mx + b → b = y - mx
"""
y_intercept = y_of_known_point - slope_of_line * x_of_known_point
return y_intercept
# Example: line through (1, 3) and (4, 9)
m = compute_slope_from_two_points(1, 3, 4, 9)
print(f"Slope: {m}") # 2.0
b = compute_y_intercept_from_point_and_slope(1, 3, m)
print(f"y-intercept: {b}") # 1.0
print(compute_y_from_slope_intercept(m, b, x_value=0)) # 1.0 (the y-intercept)
print(compute_y_from_slope_intercept(m, b, x_value=3)) # 7.0
```
---
## Sora 2 video prompt
```
8-second animation. A coordinate grid. A line y=mx+b appears. Two points
are marked — horizontal arrow labeled 'run = Δx' and vertical arrow labeled
'rise = Δy'. The fraction m = rise/run forms visually. The slope value m
changes: 0 (flat), 1, 3, −1 — the line tilts each time, showing the rise/run
triangle growing and shrinking. Formula y=mx+b glows throughout. Warm earth
tones, clean grid, large clear labels.
```
---
## The three forms say the same thing
```python
# These are all the same line: slope 2, y-intercept 1
# Slope-intercept: y = 2x + 1
# Point-slope (using point (0,1)): y − 1 = 2(x − 0)
# Standard form: −2x + y = 1 (or 2x − y = −1)
# Convert standard form Ax + By = C to slope-intercept:
def standard_to_slope_intercept(A, B, C):
# Ax + By = C → y = (C − Ax) / B → y = (C/B) − (A/B)x
slope = -A / B
y_intercept = C / B
return slope, y_intercept
slope, intercept = standard_to_slope_intercept(A=2, B=-1, C=-1)
print(f"y = {slope}x + {intercept}") # y = 2.0x + 1.0
```
---
## Builds on
- [Rectangle Area — Counting the Grid](/articles/66820765-0ec9-4197-8cfa-55bb81c2bf6d) — grid is the coordinate plane
## See also
- [Distance Formula — Pythagorean on a Grid](/articles/distance-formula)
- [Sine and Cosine — Waves from a Spinning Point](/articles/sin-cos)
- [Visual Tangent Without Math Libraries](/articles/79e757c8-c0af-4543-b5f0-e68ac4c39e39)
- [Math Foundations — Visual Table of Contents](/articles/d404884f-54fc-4289-b3f1-baaad2bec6b2)