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:

\[x=\frac{b_{1}c_{2} − b_{2}c_{1}}{a_{1}b_{2} − a_{2}b_{1}} \]

\[y=\frac{c_{1}a_{2} − c_{2}a_{1}}{a_{1}b_{2} − a_{2}b_{1}}\]

\[(x,y)=(\frac{b_{1}c_{2} − b_{2}c_{1}}{a_{1}b_{2} − a_{2}b_{1}}, \frac{c_{1}a_{2} − c_{2}a_{1}}{a_{1}b_{2}−a_{2}b_{1}})\]

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:

\[n(A \cup B \cap C)=n(A) + n(B) + n(C) − n(A \cap B) − n(B \cap C) − n(C \cap A) + n(A \cap B \cap C)\]

The Formula of A Intersection B

If we have 2 sets, say A and B. The formula of A intersection B will be as follows:

\[n(A \cup B)=n(A) + n(B) − n(A \cap B) \]

Where 

\[n(A \cup B)\] is the number of elements present in either set A or set B.

\[n(A \cap B)\] is the number of elements present in both set A and set B.

Problems on Point of Intersection of Two Lines Formula:

1. Find the Point of Intersection of Two Lines \[2x + 4y + 6 = 0\] and \[2x + 3y + 4 = 0\].

Ans: The two line equations are given as \[2x + 4y + 6 = 0\] and \[2x + 3y + 4 = 0\].

Comparing these two equations with \[a_{1}x+b_{1}y+c_{1} and a_{2}x+b_{2}y+c_{2}\]

We get \[a_{1}=2,b_{1}=4,c_{1}=6, a_{2}=2,b_{2}=3,c_{2}=4 \]

Point of intersection formula is given as

\[(x,y)=(\frac{b_{1}c_{2} − b_{2}c_{1}}{a_{1}b_{2} − a_{2}b_{1}}, \frac{c_{1}a_{2} − c_{2}a_{1}}{a_{1}b_{2} − a_{2}b_{1}})\]

Substituting the values of \[a_{1},b_{1},c_{1},a_{2},b_{2},c_{2}\] in the intersection formula

\[(x,y)=(\frac{4\times4 − 3\times6}{2\times3 − 2\times4},\frac{6\times2 − 4\times2}{2\times3 − 2\times4})\]

\[(x,y)=(\frac{16 − 18}{6 − 8},\frac{12 − 8}{6 − 8})\]

\[(x,y)=(\frac{-2}{-2},\frac{4}{-2})\]

\[(x,y)=(1,−2)\]

Therefore the point of intersection of two lines \[2x + 4y + 6 = 0\] and \[2x + 3y + 4 = 0\] is \[(x,y)=(1,−2)\].

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.

Example: Find the point of intersection of two lines \[2x + 4y + 2 = 0\] and \[2x + 3y + 5 = 0\] using the point of intersection formula.

Solution:

Given Straight Line Equations are:

\[2x + 4y + 2 = 0\] and \[2x + 3y + 5 = 0\]

Here,

\[a_{1} = 2, b_{1} = 4, c_{1} = 2\]

\[a_{2} = 2, b_{2} = 3, c_{2} = 5\]

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

\[x = \frac{b_{1}c_{2} - b_{2}c_{1}}{a_{1}b_{2} - a_{2}b_{1}} ; y = \frac{a_{2}c_{1} - a_{1}c_{2}}{a_{1}b_{2} - a_{2}b_{1}} \]

\[(x,y) = \frac{4\times5 - 3\times2}{2\times3 - 2\times4}, \frac{2\times2 - 2\times5}{2\times3 - 2\times4}\]

\[(x,y) = \frac{20 - 6}{6 - 8}, \frac{4 - 10}{6 -8}\]

\[(x, y) = (-7, 3) \]

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