# Point of Intersection of Two Lines – Formula and Example

Here you will learn how to find point of intersection of two lines with examples.

Let’s begin –

## How to find Point of Intersection of Two Lines

Let the equations of two lines be

$$a_1x + b_1y + c_1$$ = 0

and,  $$a_2x + b_2y + c_2$$ = 0

Suppose these two lines intersect at a point P($$x_1, y_1$$). Then, ($$x_1, y_1$$) satisfies each of the given equations.

$$\therefore$$  $$a_1x_1 + b_1y_1 + c_1$$ = 0   and   $$a_2x_1 + b_2y_1 + c_2$$ = 0

Solving these two by cross-multiplication, we get

$$x_1\over {b_1c_2 – b_2c_1}$$ = $$y_1\over {c_1a_2 – c_2a_1}$$ = $$1\over {a_1b_2 – a_2b_1}$$

$$\implies$$  $$x_1$$ = $${b_1c_2 – b_2c_1}\over {a_1b_2 – a_2b_1}$$,  $$y_1$$ = $${c_1a_2 – c_2a_1}\over {a_1b_2 – a_2b_1}$$

Hence the coordinates of the point of the point of intersection of two lines are :

( $${b_1c_2 – b_2c_1}\over {a_1b_2 – a_2b_1}$$, $${c_1a_2 – c_2a_1}\over {a_1b_2 – a_2b_1}$$)

#### Formula to find Point of Intersection :

$$x_1$$ = ( $${b_1c_2 – b_2c_1}\over {a_1b_2 – a_2b_1}$$, $$y_1$$ = $${c_1a_2 – c_2a_1}\over {a_1b_2 – a_2b_1}$$)

Note : To find the coordinates of the point of intersection of two non-parallel lines, we solve the given equations simultaneously and the values of x and y are so obtained determine the coordinates of the point of intersection.

Example : Find the coordinates of the point of intersecton of the lines 2x – y + 3 = 0 and x + 2y – 4 = 0.

Solution : Solving simultaneously the equations 2x – y + 3 = 0 and x + 2y – 4 = 0, we obtain

$$x\over {4-6}$$ = $$y\over {3+8}$$ = $$1\over {4+1}$$

$$\implies$$ $$x\over -2$$ = $$y\over 11$$ = $$1\over 5$$

$$\implies$$ x = $$-2\over 5$$ , y = $$11\over 5$$

Hence, (-2/5, 11/5) is the required point of intersection

Example : Find the coordinates of the point of intersecton of the lines x – y + 4 = 0 and x + 2y – 1 = 0.

Solution : Solving simultaneously the equations x – y + 4 = 0 and x + 2y – 1 = 0, we obtain

$$x\over {1-8}$$ = $$y\over {4+1}$$ = $$1\over {2+1}$$

$$\implies$$ $$x\over -7$$ = $$y\over 5$$ = $$1\over 3$$

$$\implies$$ x = $$-7\over 3$$ , y = $$5\over 3$$

Hence, (-7/3, 5/3) is the required point of intersection

### Related Questions

Find the equation of line joining the point (3, 5) to the point of intersection of the lines 4x + y – 1 = 0 and 7x – 3y – 35 = 0.

Find the coordinates of the point of intersecton of the lines 2x – y + 3 = 0 and x + y – 5 = 0.

Find the equation of line parallel to y-axis and drawn through the point of intersection of the lines x – 7y + 5 = 0 and 3x + y = 0.