Point-Slope Form – Writing a Line from One Point and a Gradient

The point-slope form of a line's equation is y − y₁ = m(x − x₁), where m is the gradient and (x₁, y₁) is any known point on the line. It's especially handy because it lets you write down a line's equation immediately – no separate step needed to solve for c the way gradient-intercept form requires.

Point-slope form can always be rearranged into the more familiar y = mx + c form with a little algebra, so the two forms describe exactly the same line – point-slope form is just a faster starting point when you already know a gradient and a single point.

Using Point-Slope Form

y − y₁ = m(x − x₁). Substitute the known gradient m and point (x₁, y₁) directly – no need to solve for c first.

A line has gradient 2 and passes through (3, 5). Find y when x = 7.

y − 5 = 2(x − 3). At x = 7: y − 5 = 2(7 − 3) = 8, so y = 13.

Rewrite y − 4 = 3(x − 2) in the form y = mx + c.

y − 4 = 3x − 6, so y = 3x − 6 + 4 = 3x − 2. Here m = 3 and c = −2.

Real-Life Application

Physics: writing a motion equation from a starting position and a constant rate of change.
Finance: projecting future values from a known starting point and growth rate.
Engineering: describing a ramp or incline from one measured point and its slope.

Key Takeaways

Point-slope form is y − y₁ = m(x − x₁).
It lets you write a line's equation directly from a gradient and one point.
It can always be rearranged into y = mx + c form.

Practice: Point-Slope Form

Use Point-Slope Form

Point-Slope Form – Writing a Line from One Point and a Gradient

Using Point-Slope Form

Real-Life Application

Key Takeaways

Practice: Point-Slope Form

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