# Point of Intersection of Two Lines

We will learn how to find the co-ordinates of the point of intersection of two lines.

Let the equations of two intersecting straight lines be

a$$_{1}$$ x + b$$_{1}$$y + c$$_{1}$$  = 0 ………….. (i) and

a$$_{2}$$ x + b$$_{2}$$ y + c$$_{2}$$ = 0 …….…... (ii)

Suppose the above equations of two intersecting lines intersect at P(x$$_{1}$$, y$$_{1}$$). Then (x$$_{1}$$, y$$_{1}$$) will satisfy both the equations (i) and (ii).

Therefore, a$$_{1}$$x$$_{1}$$ + b$$_{1}$$y$$_{1}$$  + c$$_{1}$$ = 0 and

a$$_{2}$$x$$_{1}$$ + b$$_{2}$$y$$_{1}$$ + c$$_{2}$$ = 0

Solving the above two equations by using the method of cross-multiplication, we get,

$$\frac{x_{1}}{b_{1}c_{2} - b_{2}c_{1}} = \frac{y_{1}}{c_{1}a_{2} - c_{2}a_{1}} = \frac{1}{a_{1}b_{2} - a_{2}b_{1}}$$

Therefore, x$$_{1}$$  = $$\frac{b_{1}c_{2} - b_{2}c_{1}}{a_{1}b_{2} - a_{2}b_{1}}$$ and

y$$_{1}$$  = $$\frac{c_{1}a_{2} - c_{2}a_{1}}{a_{1}b_{2} - a_{2}b_{1}}$$,  a$$_{1}$$b$$_{2}$$ - a$$_{2}$$b$$_{1}$$ ≠ 0

Therefore, the required co-ordinates of the point of intersection of the lines (i) and (ii) are

($$\frac{b_{1}c_{2} - b_{2}c_{1}}{a_{1}b_{2} - a_{2}b_{1}}$$, ($$\frac{c_{1}a_{2} - c_{2}a_{1}}{a_{1}b_{2} - a_{2}b_{1}}$$), a$$_{1}$$b$$_{2}$$ - a$$_{2}$$b$$_{1}$$ ≠ 0

Notes: 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 so obtained determine the coordinates of the point of intersection.

If a$$_{1}$$b$$_{2}$$ - a$$_{2}$$b$$_{1}$$ = 0 then a$$_{1}$$b$$_{2}$$ = a$$_{2}$$b$$_{1}$$

$$\frac{a_{1}}{b_{1}}$$ = $$\frac{a_{2}}{b_{2}}$$

- $$\frac{a_{1}}{b_{1}}$$ = - $$\frac{a_{2}}{b_{2}}$$  i.e., the slope of line (i) = the slope of  line  (ii)

Therefore, in this case the straight lines (i) and (ii) are parallel and hence they do not intersect at any real point.

Solved example to find the co-ordinates of the point of intersection of two given intersecting straight lines:

Find the coordinates of the point of intersection of the lines 2x - y + 3 = 0 and x + 2y - 4 = 0.

Solution:

We know that the co-ordinates of the point of intersection of the lines a$$_{1}$$ x+ b$$_{1}$$y+ c$$_{1}$$  = 0 and a$$_{2}$$ x + b$$_{2}$$ y + c$$_{2}$$ = 0 are

($$\frac{b_{1}c_{2} - b_{2}c_{1}}{a_{1}b_{2} - a_{2}b_{1}}$$, ($$\frac{c_{1}a_{2} - c_{2}a_{1}}{a_{1}b_{2} - a_{2}b_{1}}$$), a$$_{1}$$b$$_{2}$$ - a$$_{2}$$b$$_{1}$$ ≠ 0

Given equations are

2x - y + 3 = 0 …………………….. (i)

x + 2y - 4 = 0 …………………….. (ii)

Here a$$_{1}$$ = 2, b$$_{1}$$ = -1, c$$_{1}$$ = 3, a$$_{2}$$ = 1, b$$_{2}$$ = 2 and c$$_{2}$$ = -4.

($$\frac{(-1)\cdot (-4) - (2)\cdot (3)}{(2)\cdot (2) - (1)\cdot (-1)}$$, $$\frac{(3)\cdot (1) - (-4)\cdot (2)}{(2)\cdot (2) - (1)\cdot (-1)}$$)

($$\frac{4 - 6}{4 + 1}$$, $$\frac{3 + 8}{4 + 1}$$)

($$\frac{11}{5}, \frac{-2}{5}$$)

Therefore, the co-ordinates of the point of intersection of the lines 2x - y + 3 = 0 and x + 2y - 4 = 0 are ($$\frac{11}{5}, \frac{-2}{5}$$).

The Straight Line

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### What is a Triangle? | Types of Triangle | Scalene Triangle | Isosceles

Jun 17, 24 11:22 PM

A simple closed curve or a polygon formed by three line-segments (sides) is called a triangle. The above shown shapes are triangles. The symbol of a triangle is ∆. A triangle is a polygon with three s…

2. ### Interior and Exterior of an Angle | Interior Angle | Exterior Angle

Jun 16, 24 05:20 PM

Interior and exterior of an angle is explained here. The shaded portion between the arms BA and BC of the angle ABC can be extended indefinitely.

3. ### Angles | Magnitude of an Angle | Measure of an angle | Working Rules

Jun 16, 24 04:12 PM

Angles are very important in our daily life so it’s very necessary to understand about angle. Two rays meeting at a common endpoint form an angle. In the adjoining figure, two rays AB and BC are calle

4. ### What is a Polygon? | Simple Closed Curve | Triangle | Quadrilateral

Jun 16, 24 02:34 PM

What is a polygon? A simple closed curve made of three or more line-segments is called a polygon. A polygon has at least three line-segments.