 # Point of Intersection Formula

Point of intersection means the point at which two lines intersect. These two lines are represented by the equation a1x + b1y + c1= 0  and a2x + b2y + c2 = 0, respectively. Given figure illustrate the point of intersection of two lines. We can find the point of intersection of three or more lines also. By solving the two equations, we can find the solution for the point of intersection of two lines.

The formula of the point of Intersection of two lines is:

(x, y)  = [

$$\begin{array}{l}\frac{b_{1}c_{2}-b_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}}\end{array}$$
$$\begin{array}{l}\frac{a_{2}c_{1}-a_{1}c_{2}}{a_{1}b_{2}-a_{2}b_{1}}\end{array}$$
]

### Solved Example

Question: Find out the point of intersection of two lines x + 2y + 1 = 0 and 2x + 3y + 5 = 0.

Solution:

Given straight line equations are:

x + 2y + 1 = 0 and 2x + 3y + 5 = 0

Here,
a1 = 1, b1 = 2, c1 = 1

a2 = 2, b2 = 3, c2 = 5

Intersection point can be calculated using this formula,

x =

$$\begin{array}{l}\frac{b_{1}c_{2}-b_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}}\end{array}$$
; y =
$$\begin{array}{l}\frac{a_{2}c_{1}-a_{1}c_{2}}{a_{1}b_{2}-a_{2}b_{1}}\end{array}$$

(x,y) = (

$$\begin{array}{l}\frac{2\times5-3\times1}{1\times3-2\times2}\end{array}$$
,
$$\begin{array}{l}\frac{2\times1-1\times5}{1\times3-2\times2}\end{array}$$
)

(x,y) = (

$$\begin{array}{l}\frac{10-3}{3-4}\end{array}$$
,
$$\begin{array}{l}\frac{2-5}{3-4}\end{array}$$
)

(x,y) = (-7, 3)