Equation of a line through two points
Find the gradient, then use y − y₁ = m(x − x₁).
Key formulas
Steps
Compute m. Pick either point as (x₁,y₁). Expand only if the question wants general form ax+by+c=0.
Example
Through (1,2) and (3,8): m=3. Equation y−2=3(x−1) → y=3x−1.
Vertical line
If x₂=x₁, gradient is undefined. The equation is simply x = that constant.
Worked example
Worked example
Problem. Find the equation of the line through (1, 2) and (3, 8).
Solution. m = (8−2)/(3−1) = 3. Using (1,2): y − 2 = 3(x − 1) → y = 3x − 1.
Verify
Both given points must satisfy the finished equation.
Review quiz
Line through (0,4) and (2,0) is:
Practice questions
- Line through (0,4) and (2,0). Find its equation.
- Points (5,1) and (5,−3). Write the equation.
Study notes
Point-slope vs two-point form?
They are equivalent; two-point form hides m in a fraction.