Slope and the Equations of a Line

Calculus Readiness · Axiom Academy

LESSON Slope and the Equations of a Line One number measures a line's steepness — and y = mx + b and y - y_1 = m(x - x_1) are two ways of writing the same line. Pick any two points on a line and walk between them: some distance sideways (the run ) and some distance up or down (the rise ). Slope is the quotient of the two. Watch what happens as the line tilts — the run is held fixed at 3 while the rise, and with it the ratio, changes. Vertical change over horizontal change m > 0 : the line rises left to right. The bigger m is, the steeper the climb. m < 0 : the line falls left to right. The rise is negative while the run stays positive. m = 0 : a horizontal line, y = c . No rise at all, so the quotient is 0 . A vertical line x = a has run 0 , and you cannot divide by zero. Its slope does not exist — it is not "infinite". Given (x_1, y_1) and (x_2, y_2) , the run is x_2 - x_1 and the rise is y_2 - y_1 . Divide and you have the slope. The animation holds one line still and slides the pair of sample points along it: the triangle grows and shrinks, but the quotient does not move. Subtract in the same order, top and bottom The slope of the line through (1, 2) and (5, 10) . Reversing both subtractions gives — the same answer. What you must not do is mix the orders: is the slope of a different line. Any two rise/run triangles on the same line are similar, so their corresponding sides are in the same ratio.

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