Where two lines meet is where both equations are true – the break-even point of costs and revenue, the meeting point of two travellers, or the solution of a pair of linear equations.
How to use the Intersection of Two Lines Calculator
- Equation form
- Line 1: a₁ (slope) or A₁
- Line 1: b₁ (intercept) or B₁
- Line 1: C₁
- Line 2: a₂ (slope) or A₂
- Line 2: b₂ (intercept) or B₂
- Line 2: C₂
Results update as you type, and everything is calculated in your browser.
How it works
The calculator solves the two equations simultaneously. Parallel lines (equal slopes) do not intersect; identical lines meet everywhere.
Inputs explained
| Field | What to enter |
|---|---|
| Equation form | Choose one: Slope-intercept: y = a·x + b, General: A·x + B·y = C |
| Line 1: a₁ (slope) or A₁ | A number |
| Line 1: b₁ (intercept) or B₁ | A number |
| Line 1: C₁ | A number |
| Line 2: a₂ (slope) or A₂ | A number |
| Line 2: b₂ (intercept) or B₂ | A number |
| Line 2: C₂ | A number |
Example values
How the main result changes when Line 1: a₁ changes and everything else stays at the example values above:
| Line 1: a₁ (slope) or A₁ | Intersection x₀ |
|---|---|
| 1 | 3.3333 |
| 1.5 | 2.5 |
| 2 | 2 |
| 2.5 | 1.6667 |
| 3 | 1.4286 |
| 4 | 1.1111 |
Formula
Slope-intercept: x₀ = (b₂ − b₁)/(a₁ − a₂), y₀ = a₁x₀ + b₁
General: x₀ = (C₁B₂ − C₂B₁)/(A₁B₂ − A₂B₁), y₀ = (A₁C₂ − A₂C₁)/(A₁B₂ − A₂B₁)
Worked example
Inputs — Equation form: Slope-intercept: y = a·x + b; Line 1: a₁ (slope) or A₁: 2; Line 1: b₁ (intercept) or B₁: -1; Line 2: a₂ (slope) or A₂: -0.5; Line 2: b₂ (intercept) or B₂: 4.
Results — Intersection x₀ = 2; Intersection y₀ = 3; Angle between the lines = 90 °.
Key terms
- Intersection
- The point common to two lines.
Frequently asked questions
What if the lines are parallel?
They never meet; the formulas divide by zero.
How do I convert y = ax + b to general form?
−a·x + y = b, so A = −a, B = 1, C = b.
Can two lines intersect at more than one point?
Only if they are the same line.
What about lines in 3D?
They may be skew – neither parallel nor intersecting.