Find the slope, y-intercept, and line equation through any two coordinate points.
Slope measures the rate of change of a line — how much the y-value changes for each one-unit increase in x. A slope of 3 means y increases by 3 for every 1-unit rightward movement. A slope of -0.5 means y decreases by 0.5 for each 1-unit increase in x. The formula (y₂-y₁)/(x₂-x₁) is often described as "rise over run" — the vertical change (rise) divided by the horizontal change (run) between any two points on the line.
Slope is fundamental to understanding linear relationships in any context. In economics, a demand curve's slope tells you how much quantity demanded changes per dollar of price increase. In physics, velocity is the slope of a position-vs-time graph. In statistics, the slope of a regression line tells you how much the dependent variable changes per unit increase in the independent variable. Any linear relationship can be characterized by its slope.
A positive slope means the line goes up from left to right — the variables increase together, suggesting a positive relationship. A negative slope means the line goes down left to right — as one variable increases, the other decreases. A slope of zero is a horizontal line — y doesn't change regardless of x, meaning no relationship between the variables. An undefined slope occurs for a vertical line — x doesn't change but y does, which requires dividing by zero in the formula.
The magnitude matters too: a slope of 10 represents a steeper line than a slope of 0.5. Steeper slopes mean greater rates of change. In business, a revenue curve with slope 5 (revenue increases by $5 per unit sold) is more profitable per unit than one with slope 2 — the difference in slope is the difference in gross margin per unit.
Once you have the slope m and a point (x₁, y₁), you can write the full line equation in point-slope form: y - y₁ = m(x - x₁). Rearranging gives slope-intercept form: y = mx + b, where b is the y-intercept (the value of y when x = 0). The y-intercept is the "starting value" when x is zero — in a cost model, it might represent fixed costs; in a physics problem, initial position.
Two special cases: parallel lines have identical slopes (same rate of change, different starting points). Perpendicular lines have slopes that are negative reciprocals of each other — if one line has slope 3, the perpendicular line has slope -1/3. This relationship is used in geometry, computer graphics, and engineering to construct right angles.
Calculus extends the concept of slope from straight lines to curves. The derivative of a function at a point is the slope of the tangent line at that point — the instantaneous rate of change. For a curve like y = x², the slope varies at every point: at x=2 the slope is 4, at x=5 the slope is 10. Calculus gives us the tools to compute these varying slopes, which is why derivatives are so powerful for optimization (finding where slope equals zero, identifying maxima and minima) and physics (velocity is the derivative of position, acceleration is the derivative of velocity).