Point-Slope Form

Algebra 1 · Axiom Academy

Discover a powerful form of a line that builds directly from the definition of slope itself. 1. Foundation: The Slope Formula Everything starts with the definition of slope: the rise over the run between any two points on a line. Given two points (x_1, y_1) and (x_2, y_2) , the slope m is the same no matter which two points you pick. What if we know one point (x_1, y_1) and the slope m , but not a second point? We let a generic point (x, y) stand for any other point on the line. The slope between (x_1, y_1) and (x, y) must still equal m — so we just clear the fraction. Point-slope form is your best friend when you are handed a slope and one point and asked to write the line. You don't need the y -intercept first — just drop the numbers straight into the form. Write the equation of the line with slope m = 3 that passes through the point (4, 2) . Substituting x_1 = 4 , y_1 = 2 , m = 3 gives 4. Converting to Slope-Intercept Form Often you want the answer in slope-intercept form, y = mx + b . From y - 2 = 3(x - 4) it's a two-step conversion: distribute the slope, then isolate y . 5. Converting to Standard Form You can also write the line in standard form, Ax + By = C , where A , B , C are integers and A is non-negative. Starting from y = 3x - 10 , gather the x and y terms on one side and the constant on the other. You've seen that point-slope form is just the slope formula rearranged — and how to use it to write a line and convert it into every other form.

This is the written version of the interactive lesson above. See the full Algebra 1 course.