5. The Coordinate Plane

The coordinate plane is a map made of two number lines: a horizontal x-axis and a vertical y-axis crossing at the origin \((0,0)\). Any point is named by an ordered pair \((x, y)\) — go right/left \(x\), then up/down \(y\).

Slope measures steepness: 'rise over run,' how much a line goes up for each step to the right.
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1}. \]
Between \((1, 2)\) and \((4, 8)\): \(\text{slope} = \frac{8-2}{4-1} = \frac{6}{3} = 2\). The line rises 2 for every 1 across.

Distance between two points uses the Pythagorean theorem in disguise:
\[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}. \]

💡 Tip: a positive slope climbs left-to-right; a negative slope falls; a horizontal line has slope \(0\).