LINEAR EQUATIONS

Visualizing relationships between two variables. Slope, intercepts, and the geometry of the infinite line.

01 // The Coordinate Plane

A 2D space defined by two perpendicular number lines: the horizontal X-Axis and vertical Y-Axis. They intersect perfectly in the center at the Origin (0,0). Every point in this space represents an exact mathematical pairing.

X-AXIS

Independent Var

Y-AXIS

Dependent Var

Q1 (+,+)

Q2 (-,+)

Q3 (-,-)

Q4 (+,-)

02 // Slope (m) & Rise/Run

The Rate of Change

Slope measures the steepness and direction of a line. It is the ratio of the vertical change between two points (the rise) to the horizontal change (the run). Drag the points in the lab below to see it in action!

y₂ - y₁x₂ - x₁

Δ RiseΔ Run

Two-Point Constructor

Point 1 (x₁, y₁)(-4, -2)

X-Axis

Y-Axis

Point 2 (x₂, y₂)(4, 2)

X-Axis

Y-Axis

Rise (Δy)4

Run (Δx)8

Equation:y = 0.50x + 0

15px / Unit

03 // Equation Forms

SLOPEThe constant rate of change.

Y-INTERCEPTWhere the line crosses the y-axis (0, b).

Default

Slope-Intercept

y = mx + b

The gold standard. Best for graphing directly because it explicitly states your starting point and your path.

Builder

Point-Slope

y - y₁ = m(x - x₁)

The constructor. Best for building an equation when you only have a single random point and a slope.

Formal

Standard Form

Ax + By = C

The organizer. Best for finding the exact X and Y intercepts quickly by plugging in zeros.

Practice Arena // Coming Soon

Dynamic Problem Sets

The question engine is currently offline while we aggregate vocabulary and construct the generation architecture. Check back soon for infinite practice problems!