Equation, Table, and Graph of a Line

A line is just a constant rate of change. Because the rate (slope) never changes, any two points on the line determine the whole thing: slope is rise over run between them, and one point lets you finish the equation. That is why two points are always enough, no matter which two you pick.

Standard form, like 2x + 3y = 12, hides the slope and intercepts inside the coefficients. Rewrite it in slope-intercept form, y = mx + b, and both become visible: b is where the line crosses the vertical axis, m tells you how many units y moves for every one unit x moves. That makes graphing fast — plot the intercept, then use the slope to step to a second point. Building a table works the same way: pick easy x-values, substitute, and solve for y, undoing whatever is multiplying or adding to it.

This topic is the hub connecting the algebra of linear equations to their graphs, and it feeds directly into systems of equations, where you find where two lines meet, and into inequalities, where the line becomes a boundary. Once you can move freely between equation, table, and graph, those topics are just this one applied twice or shaded on one side.