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Point of Intersection Calculator
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Intersection of two lines is a point as shown below.

Here, P is the point of intersection of lines 1 and 2.

Deriving the formula for finding the point of intersection of two lines:

Let a1x + b1y + c1 = 0 ......... (1) and a2x + b2y + c2 = 0 ......... (2) be two intersecting lines.

Let (x, y) be the intersecting point of  lines (1) and (2), then the point (x, y) should lie on both the lines represented by (1) and (2). i.e. the point (x, y)satisfies both the equations (1) and (2).

Solving the equations (1) and (2), we get the intersection point of two line (x, y)

$\frac{x}{(b_{1}c_{2} - b_{2}c_{1})}$ = $\frac{y}{(a_{2}c_{1} - a_{1}c_{2})}$ = $\frac{1}{(a_{1}b_{2} - a_{2}b_{1})}$

So, x = $\frac{(b_{1}c_{2} - b_{2}c_{1})}{(a_{1}b_{2} - a_{2}b_{1})}$

y = $\frac{(a_{2}c_{1} - a_{1}c_{2})}{(a_{1}b_{2} - a_{2}b_{1})}$

Therefore, point of intersection (x, y) = [ $\frac{(b_{1}c_{2} - b_{2}c_{1})}{(a_{1}b_{2} - a_{2}b_{1})}$, $\frac{(a_{2}c_{1} - a_{1}c_{2})}{(a_{1}b_{2} - a_{2}b_{1})}$ ]

Point of Intersection Calculator will help us to find the point the point of intersection for a given system of equations.

## Steps for Point of Intersection Calculator

Step 1 :

Observe the given system of equations.

Step 2 :

Apply the formula:

point of intersection (x, y) = [ $\frac{(b_{1}c_{2} - b_{2}c_{1})}{(a_{1}b_{2} - a_{2}b_{1})}$, $\frac{(a_{2}c_{1} - a_{1}c_{2})}{(a_{1}b_{2} - a_{2}b_{1})}$ ]

## Problems on Point of Intersection Calculator

1. ### Find the point of intersection of the lines:x + y = 12x + 3y = 5

Step 1 :

Given system of equations:

x + y = 1

2x + 3y = 5

So, a1 = 1, b1 = 1 and c1 = -1

a2 = 2, b2 = 3 and c2 = -5

Step 2 :

point of intersection (x, y) = [ $\frac{(b_{1}c_{2} - b_{2}c_{1})}{(a_{1}b_{2} - a_{2}b_{1})}$, $\frac{(a_{2}c_{1} - a_{1}c_{2})}{(a_{1}b_{2} - a_{2}b_{1})}$ ]

point of intersection (x, y) = [ $\frac{((1)(-5) - (3)(-1))}{((1)(3) - (2)(1))}$, $\frac{((-2)(1) - (-1)(5))}{((1)(3) - (2)(1))}$]

point of intersection (x, y) = (-2, 3)

(2, -3)

2. ### Find the point of intersection of the lines:x + y = -12x + 3y = -5

Step 1 :

Given system of equations:

x + y = -1

2x + 3y = -5

So, a1 = 1, b1 = 1 and c1 = 1

a2 = 2, b2 = 3 and c2 = 5

Step 2 :

point of intersection (x, y) = [ $\frac{(b_{1}c_{2} - b_{2}c_{1})}{(a_{1}b_{2} - a_{2}b_{1})}$, $\frac{(a_{2}c_{1} - a_{1}c_{2})}{(a_{1}b_{2} - a_{2}b_{1})}$ ]

point of intersection (x, y) = [ $\frac{((1)(5) - (3)(1))}{((1)(3) - (2)(1))}$, $\frac{((2)(1) - (1)(5))}{((1)(3) - (2)(1))}$]

point of intersection (x, y) = (2, -3)