Point of intersection means the point at which two lines intersect. These two lines are represented by the equation a_{1}x+ b_{1}y + c_{1}= 0 and a_{2}x+ b_{2}y + c_{2} = 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:

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### 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,

a_{1} = 1, b_{1} = 2, c_{1} = 1

a_{2} = 2, b_{2} = 3, c_{2} = 5

Intersection point can be calculated using this formula,

## Point of Intersection Formula

The point of intersection formula is used to find the point of intersection of two lines, meaning the meeting point of two lines. These two lines can be represented by the equation a1x+b1y+c1=0 and a2x+b2y+c2=0, respectively. It is possible to find the point of intersection of three or more lines. By solving the two equations, we can find the solution for the point of intersection of two lines. Let us learn about the point of intersection formula with a few solved examples.

## What Is Point of Intersection Formula?

Consider 2 straight lines a1x+b1y+c1=0, and a2x+b2y+c2=0 which are intersecting at point (x,y) as shown in figure. So, we have to find a line intersection formula to find these points of intersection (x,y). These points satisfy both the equations. By solving these two equations we can find the point where two lines intersect. The formula of the point of Intersection of two lines is:

Let us have a look at a few solved examples to understand the point of intersection formula better.

## Solved Examples Using Point of Intersection Formula

Example 1: Using point of intersection formula, find the point of intersection of two lines 2x + 4y + 2 = 0 and 2x + 3y + 5 = 0.

**Solution:**

Given straight line equations are:

2x + 4y + 2 = 0 and 2x + 3y + 5 = 0

Here,

a1=2,b1=4,c1=2

a2=2,b2=3,c2=5

Intersection point can be calculated using the point of intersection formula,

**Answer**: **Thus, the point of intersection is (-7,3).**

Example 2: What will be the point of Intersection of the given two lines 2x+4y+6=0

and 6x+9y+12=0

**Solution**:

The two line equations are given as 2x+4y+6=0 and 6x+9y+12=0.

Comparing these two equations with a1x+b1y+c1anda2x+b2y+c2

We get a1=2,b1=4,c1=6

a2=6,b2=9,c2=12

Point of intersection formula is given as

## Point of Intersection Formula – Two Lines Formula and Solved Problems

In mathematics, we refer to the point of intersection where a point meets two lines or curves. The intersection of lines may be an empty set, a point, or a line in Euclidean geometry. A required criterion for two lines to intersect is that they should be in the same plane and are not skew lines. The intersection formula gives the point at which these lines are meeting.

**Point of Intersection of Two Lines Formula**

Consider 2 straight lines

*a*1*x*+*b*1*y*+*c*1 and

*a*2*x*+*b*2*y*+*c*2

which are intersecting at point (x,y) as shown in figure. So, we have to find a line intersection formula to find these points of intersection (x,y). These points satisfy both the equations.

By solving these two equations we can find the intersection of two lines formula.

The formula for the point of intersection of two lines will be as follows:

So, this is the point of intersection formula of 2 lines when intersecting at one point.

**A Intersection B Intersection C Formula**

If we have 3 sets A,B and C. A intersection B intersection C formula will be as follows:

The point of intersection formula is used to find the meeting point of two lines, also known as the point of intersection. The equation can be used to represent these two lines

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

and

*a*2*x*+*b*2*y*+*c*2=0

respectively. The point of intersection of three or more lines can be found. We can discover the solution for the point of intersection of two lines by solving the two equations. Let’s look at some solved examples of the point of intersection formula.

Two straight lines will intersect at a point if they are not parallel. The point of intersection is the meeting point of two straight lines.

If two crossing straight lines have the same equations, the intersection point can be found by solving both equations at the same time.

On a two-dimensional graph, straight lines intersect at just one location, which is described by a single set of display style x-coordinates and display style y-coordinates. You know the display style x-coordinates and display style y-coordinates must fulfil both equations since both lines pass through that location.

**Intersection Point**

Have you ever been driving and come across a traffic sign like this?

This is an intersection traffic sign, and it means you’re approaching a place where two roads connect. Two lines cross and meet in the centre of the intersection traffic sign, as you can see. This is where their paths cross. The intersection of two lines or curves is referred to as a point of intersection in mathematics.

The intersection of two curves is noteworthy because it is the point at which the two curves have the same value. This can come in handy in a variety of situations. Let’s imagine we’re dealing with an equation that represents a company’s revenue and another that represents the company’s expense. The point of intersection of the curves corresponding to these two equations is where revenue equals cost; this is the company’s breakeven point.

**Application for a Map of Points of Intersection**

**Review**

- A point of intersection is the meeting point of two lines or curves.
- By graphing the curves on the same graph and finding their points of intersection, we can discover a point of junction graphically.
- The following steps can be used to find a point of intersection algebraically:
- For one of the variables, call it y, and solve each equation.
- Set the equations for y discovered in the first step to the same value, then solve for the other variable, which we’ll call x. This is the x-value of the junction point.
- In any of the original equations, plug in the x-value of the site of intersection and solve for y. This is the point of intersection y-variable.

**Question 1: Find the point of intersection of line 3x + 4y + 5 = 0, 2x + 5y +7 = 0.**

**Solution:**

The point of intersection of two lines is given by :

(x, y) = ((b1*c2−b2*c1)/(a1*b2−a2*b1), (c1*a2−c2*a1)/(a1*b2−a2*b1))

a1 = 3, b1 = 4, c1 = 5

a2 = 2, b2 = 5, c2 = 7

(x,y) = ((28-25)/(15-8), (10-21)/(15-8))

(x,y) = (3/7,-11/7)

**Question 2: Find the point of intersection of line 9x + 3y + 3 = 0, 4x + 5y + 6 = 0.**

**Solution:**

The point of intersection of two lines is given by :

(x,y) = ((b1*c2−b2*c1)/(a1*b2−a2*b1), (c1*a2−c2*a1)/(a1*b2−a2*b1))

a1 = 9, b1 = 3, c1 = 3

a2 = 4, b2 = 5, c2 = 6

(x, y) = ((18-15)/(45-15), (54-12)/(45-15))

(x, y) = (1/10, 7/5)

**Question 3: Check if the two lines are parallel or not 2x + 4y + 6 = 0, 4x + 8y + 6 = 0**

**Solution:**

To check whether the lines are parallel or not we need to check a1/b1 = a2/b2

a1 = 2, b1 = 4

a2 = 4, b2 = 8

2/4 = 4/8

1/2 = 1/2

Since the condition is satisfied the lines are parallel and can’t intersect each other.

**Question 4: Check if the two lines are parallel or not 3x + 4y + 8 = 0, 4x + 8y + 6 = 0**

**Solution:**

To check whether the lines are parallel or not we need to check a1/b1 = a2/b2

a1 = 3, b1 = 4

a2 = 4, b2 = 8

3/4 is not equal to 4/8

Since the condition is not satisfied the lines are not parallel.

**Question 5: Check whether the point (3, 5) is point of intersection of lines 2x + 3y – 21 = 0, x + 2y – 13 = 0.**

**Solution:**

A point to be a point of intersection it should satisfy both the lines.

Substituting (x,y) = (3,5) in both the lines

Check for equation 1: 2*3 + 3*5 – 21 =0 —-> satisfied

Check for equation 2: 3 + 2* 5 -13 =0 —-> satisfied

Since both the equations are satisfied it is a point of intersection of both the lines.

**Question 6: Check whether the point (2, 5) is point of intersection of lines x + 3y – 17 = 0, x + y – 13 = 0**

**Solution:**

A point to be a point of intersection it should satisfy both the lines.

Substituting (x,y) = (2,5) in both the lines

Check for equation 1: 2+ 3*5 – 17 =0 —-> satisfied

Check for equation 2: 7 -13 = -6 —>not satisfied

Since both the equations are not satisfied it is not a point of intersection of both the lines.

**Question 7: Find the point of intersection of lines x = -2 and 3x + y + 4 = 0**

**Solution:**

On substituting x = -2 in 3x + y + 4 = 0

-6 + y + 4 = 0;

y = 2;

So the point of intersection is (x,y) = (-2,2)

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