Geometry
Gradient-Intercept Form – Building y = mx + c from Points
Knowing that a line follows y = mx + c is only half the story – the real skill is building that equation from raw information, like two points on the line. The gradient formula, m = (y₂ − y₁) ÷ (x₂ − x₁), lets you calculate m directly from any two points, and once you have m plus a single point, you can solve for c by substitution.
This two-step process – find m, then find c – is one of the most common tasks in coordinate geometry, and it's the exact method you'll use again and again throughout the rest of this section.
Building the Equation Step by Step
Step 1: find the gradient m = (y₂ − y₁) ÷ (x₂ − x₁). Step 2: substitute m and one known point into y = mx + c, then solve for c.
m = (13 − 4) ÷ (4 − 1) = 9 ÷ 3 = 3.
Substitute into y = 3x + c: 4 = 3(1) + c, so 4 = 3 + c, giving c = 1. The equation is y = 3x + 1.
Real-Life Application
- Data trends: finding the equation of a trend line from two data points.
- Construction: calculating a ramp's slope from two measured heights.
- Finance: building a cost formula from two known price points.
Key Takeaways
- The gradient formula m = (y₂ − y₁) ÷ (x₂ − x₁) works from any two points.
- Once m is known, substitute one point into y = mx + c to solve for c.
- Together these two steps fully reconstruct a line's equation.