Geometry Lab reference
Slope & line equations
Describe a direction, then locate the line.
m = (y₂ − y₁) / (x₂ − x₁)
The idea
Slope is vertical change divided by horizontal change. Positive slope rises from left to right; negative slope falls. Once slope m is known, use a point to find b in y = mx + b, or write the point-slope form y − y₁ = m(x − x₁).
Work through an example
The line through (1, 2) and (4, 8) has slope (8 − 2)/(4 − 1) = 2. Substituting (1, 2) gives 2 = 2 × 1 + b, so b = 0 and the equation is y = 2x.
What to watch for
Vertical lines have undefined slope and equations such as x = 3. Horizontal lines have slope zero. Two identical points do not determine a unique line.
Make the formula move.
Explore this idea with real inputs.