LESSON 20
SLOPE & LINEAR EQUATIONS
The Mathematics of Straight Lines
What Makes a Line?
A linear equation is an equation whose graph is a straight line. Every straight line has two key characteristics: its slope (how steep it is) and its y-intercept (where it crosses the y-axis). Together, these two values completely describe any non-vertical line on the coordinate plane.
Slope and linear equations are among the most widely used tools in mathematics. They model constant rates of change — speed, cost per unit, temperature change over time, and countless other real-world relationships. Mastering them opens the door to algebra, statistics, physics, economics, and data science.
Understanding Slope
Slope measures the steepness and direction of a line. It is defined as rise over run — the vertical change divided by the horizontal change between any two points on the line. The formula is m = (y₂ − y₁) / (x₂ − x₁). A positive slope rises from left to right. A negative slope falls from left to right. A slope of zero is a horizontal line. An undefined slope is a vertical line.
Slope-Intercept Form
The most common way to write a linear equation is slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept (the value of y when x = 0). This form makes it easy to graph a line — start at the y-intercept on the y-axis, then use the slope to find additional points by moving rise units up (or down) and run units right.
Writing Equations of Lines
Given a slope and a point, use point-slope form: y − y₁ = m(x − x₁), then simplify to slope-intercept form. Given two points, first calculate the slope using the slope formula, then substitute one point into point-slope form. Two lines are parallel if they have the same slope (m₁ = m₂). Two lines are perpendicular if their slopes are negative reciprocals (m₁ × m₂ = −1).
Standard Form and Intercepts
Standard form of a linear equation is Ax + By = C, where A, B, and C are integers and A is positive. To find the x-intercept, set y = 0 and solve for x. To find the y-intercept, set x = 0 and solve for y. Plotting both intercepts and drawing a line through them is a quick way to graph any linear equation in standard form.
Key Facts
The word 'slope' comes from the Latin 'clivus' meaning hill. In many countries, road signs show slope as a percentage — a 10% grade means the road rises 10 feet for every 100 feet of horizontal distance.
The slope of a position-vs-time graph gives speed. The slope of a speed-vs-time graph gives acceleration. Slope is the mathematical language of rates of change across all of science.
A line with slope m = 1 makes a 45° angle with the x-axis. Steeper lines have slopes greater than 1; shallower lines have slopes between 0 and 1. The slope equals the tangent of the angle the line makes with the horizontal.
Linear regression — used in statistics and machine learning — finds the single straight line that best fits a cloud of data points by minimizing the total squared distance from each point to the line.