This page is all about the intersection of two lines.
In this minilesson, we will learn in detail about, How to find the point of intersection of two lines?
We all are familiar with twodimensional coordinate geometry from the earlier classes.
Primarily, it is a combination of algebra and geometry.
The renowned French philosopher and mathematician, Rene Descartes, published a book, La Géométry in 1637. In this book, he introduced the world with the systematic study of geometry from the use of algebra.
"The intersection of two lines" is an introductory topic under twodimensional coordinate geometry.
Come, let us learn in detail about how to find the point of intersection of two lines.
In this lesson, we will also learn about intersecting lines examples, the point of intersection formula, and the intersection point calculator.
Lesson Plan
What Does Intersection of Two Lines Mean?
Intersecting Lines Definition
When two lines share exactly one common point, they are called intersecting lines.
The intersecting lines share a common point. And, this common point that exists on all intersecting lines is called the point of intersection.
Here, lines \(A\) and \(B\) intersect at point \(O\), which is the point of intersection.
Examples In Real Life
Common examples of intersecting lines in real life include a pair of scissors, a folding chair, a road cross, signboard, etc.
Let's consider the following case.
We are given two lines, \({L_1}\) and \({L_2}\) , and we are required to find the point of intersection and the angle of intersection.
Evaluating the point of intersection involves solving two simultaneous linear equations.
Let the equations of the two lines be (written in the general form):
\[\begin{array}{l}{a_1}x + {b_1}y + {c_1} = 0\\{a_2}x + {b_2}y + {c_2} = 0\end{array}\]
Now, let the point of intersection be \(\left( {{x_0},{y_0}} \right)\). Thus,
\[\begin{array}{l}{a_1}{x_0} + {b_1}{y_0} + {c_1} = 0\\{a_2}{x_0} + {b_2}{y_0} + {c_2} = 0\end{array}\]
This system can be solved using Cramer’s rule to get:
\[\frac{{{x_0}}}{{{b_1}{c_2}  {b_2}{c_1}}} = \frac{{  {y_0}}}{{{a_1}{c_2}  {a_2}{c_1}}} = \frac{1}{{{a_1}{b_2}  {a_2}{b_1}}}\] 
From this relation, we can obtain the point of intersection \(\left( {{x_0},{y_0}} \right)\) as
\[\left( {{x_0},{y_0}} \right) = \left( {\frac{{{b_1}{c_2}  {b_2}{c_1}}}{{{a_1}{b_2}  {a_2}{b_1}}},\frac{{{c_1}{a_2}  {c_2}{a_1}}}{{{a_1}{b_2}  {a_2}{b_1}}}} \right)\] 
Intersection Point Calculator
To obtain the angle of intersection between these two lines, consider the figure shown above.
The equations of the two lines in slopeintercept form are:
\[\begin{align}&y = \left( {  \frac{{{a_1}}}{{{b_1}}}} \right)x + \left( {\frac{{{c_1}}}{{{b_1}}}} \right) = {m_1}x + {C_1}\\&y = \left( {  \frac{{{a_2}}}{{{b_2}}}} \right)x + \left( {\frac{{{c_2}}}{{{b_2}}}} \right) = {m_2}x + {C_2}\end{align}\]
Note in the figure above that \(\theta = {\theta _2}  {\theta _1}\), and thus
\[\begin{align}&\tan \theta = \tan \left( {{\theta _2}  {\theta _1}} \right) = \frac{{\tan {\theta _2}  \tan {\theta _1}}}{{1 + \tan {\theta _1}\tan {\theta _2}}}\\&\qquad\qquad\qquad\qquad\;\;= \frac{{{m_2}  {m_1}}}{{1 + {m_1}{m_2}}}\end{align}\] 
Conventionally, we would be interested only in the acute angle between the two lines and thus, we have to have \(\tan \theta \) as a positive quantity.
So in the expression above, if the expression \(\frac{{{m_2}  {m_1}}}{{1 + {m_1}{m_2}}}\) turns out to be negative, this would be the tangent of the obtuse angle between the two lines; thus, to get the acute angle between the two lines, we use the magnitude of this expression.
Therefore, the acute angle \(\theta \) between the two lines is
\[\theta = {\tan ^{  1}}\left {\frac{{{m_2}  {m_1}}}{{1 + {m_1}{m_2}}}} \right\] 
From this relation, we can easily deduce the conditions on \({m_1}\) and \({m_2}\) such that the two lines \({L_1}\) and \({L_2}\) are parallel or perpendicular.
Conditions for Two Lines to be Parallel or Perpendicular
If the lines are parallel, \(\theta = 0\) and \({m_1} = {m_2}\), which is obvious since parallel lines must have the same slope.
For the two lines to be perpendicular lines, \(\theta = \frac{\pi }{2}\), so that \(\cot \theta = 0\); this can happen if \(1 + {m_1}{m_2} = 0\) or \({m_1}{m_2} =  1\).
If the lines \({L_1}\) and \({L_2}\) are in the general form \(ax + by + c = 0\), the slope of this line is \(m =  \frac{a}{b}\).
Condition for Two Lines to be Parallel
Thus, the condition for \({L_1}\) and \({L_2}\) to be parallel is:
\[{m_1} = {m_2}\, \Rightarrow \,  \frac{{{a_1}}}{{{b_1}}} =  \frac{{{a_2}}}{{{b_2}}}\, \Rightarrow \,\frac{{{a_1}}}{{{b_1}}} = \frac{{{a_2}}}{{{b_2}}}\] 
Example
The line \({L_1}:x  2y + 1 = 0\) is parallel to the line \({L_2}:x  2y  3 = 0\) because the slope of both the lines is \(m = \frac{1}{2}\).
Condition for Two Lines to be Perpendicular
The condition for \({L_1}\) and \({L_2}\) to be perpendicular is:
\[\begin{align}&{m_1}{m_2} =  1\, \Rightarrow \,\left( {  \frac{{{a_1}}}{{{b_1}}}} \right)\left( {  \frac{{{a_2}}}{{{b_2}}}} \right) =  1\,\\ &\qquad\qquad\;\;\;\; \Rightarrow \,\,{a_1}{a_2} + {b_1}{b_2} = 0\end{align}\] 
Example
The line \({L_1}:x + y = 1\) is perpendicular to the line \({L_2}:x  y = 1\) because the slope of \({L_1}\) is \(  1\) while the slope of \({L_2}\) is 1.
Properties of Intersecting Lines
 The intersecting lines (two or more) always meet at a single point.
 The intersecting lines can cross each other at any angle. This angle formed is always greater than \(0^{\circ}\) and less than \(180^{\circ}\).
 Two intersecting lines form a pair of vertical angles. The vertical angles are opposite angles with a common vertex (which is the point of intersection).
Here,\(\angle a\) and \(\angle c\) are vertical angles and are equal.
Also, \(\angle b\) and \(\angle d\) are vertical angles and equal to each other.
\(\angle a+\angle d\) = straight angle = \(180^{\circ}\)
Challenging Questions
 Reduce the following equations into slopeintercept form
(i) \(x+7y=0\) (ii) \(3x+2y12\) 
If the angle between two lines is \(\frac{\pi}{4}\) and slope of one the lines is \(\frac{1}{2}\), find the slope of the other line.
Solved Examples
Example 1 
Find the point of intersection and the angle of intersection for the following two lines:
\[\begin{array}{l}x  2y + 3 = 0\\3x  4y + 5 = 0\end{array}\]
Solution
We use Cramer’s rule to find the point of intersection:
\[\begin{align}&\frac{x}{{  10  \left( {  12} \right)}} = \frac{y}{{5 9 }} = \frac{1}{{  4  \left( {  6} \right)}}\\&\Rightarrow \,\,\,\frac{x}{2} = \frac{y}{4} = \frac{1}{2}\\&\Rightarrow \,\,\,x = 1,\,\,\,y = 2\end{align}\]
Also, you can check your answer with the help of cuemath's Intersection Point Calculator
Now, the slopes of the two lines are:
\[{m_1} = \frac{1}{2},\,\,\,{m_2} = \frac{3}{4}\]
If \(\theta \) is the acute angle of intersection between the two lines, we have:
\[\begin{align}&\tan \theta = \left {\frac{{{m_2}  {m_1}}}{{1 + {m_1}{m_2}}}} \right = \left {\frac{{\frac{3}{4}  \frac{1}{2}}}{{1 + \frac{3}{8}}}} \right = \frac{2}{{11}}\\&\Rightarrow \,\,\,\theta = {\tan ^{  1}}\left( {\frac{2}{{11}}} \right) \approx {10.3^\circ}\end{align}\]
\(\therefore\) Point of intersection is \((1,2)\). 
Example 2 
Find the equation of a line perpendicular to the line \(x  2y + 3 = 0\) and passing through the point\((1, \;2)\)
Solution
Given line \(x  2y + 3 = 0\) can be written as
\[{y} = \frac{1}{2}x\,\,\, + \frac{3}{2}\]
Slope of the line \((1)\) is \({m_1}= \frac{1}{2}\).
Therefore, slope of the line perpendicular to line \((1)\) is
\[{m_2} = \frac{1}{{m_1}},\,\,\, = 2\]
Equation of a line perpendicular to the line \(x  2y + 3 = 0\) and passing through the point\((1, \;2)\) is
\[y(2) = 2(x1),\,\,\, \,\,\, \text{or} \,\,\,y= 2x\]
Which is the required equation
\(\therefore\) Equation of the required line is \(y= 2x\) 

An acute angle \(\theta \) between lines \(L_1\) and \(L_2\) with slopes \(m_1\) and \(m_2\) is given by
\[\theta = {\tan ^{  1}}\left {\frac{{{m_2}  {m_1}}}{{1 + {m_1}{m_2}}}} \right\]

If the lines \({L_1}\) and \({L_2}\) are given in the general form \(ax + by + c = 0\), the slope of this line is \(m =  \frac{a}{b}\) .
The condition for two lines \({L_1}\) and \({L_2}\) to be parallel is:
\({m_1} = {m_2}\)
The condition for two lines \({L_1}\) and \({L_2}\) to be perpendicular is:
\({m_1}{m_2} =  1\)
Interactive Questions
Here are a few activities for you to practice.
Select/Type your answer and click the "Check Answer" button to see the result.
Let's Summarize
The minilesson targeted the fascinating concept of "The intersection of two lines." The math journey around "The intersection of two lines" starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.
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Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.
Frequently Asked Questions (FAQs)
1. How do I find the point of intersection of two lines?
Here's the summary of our methods:
Get the two equations for the lines into slopeintercept form. That is, have them in this form: y = mx + b.
Set the two equations for y equal to each other.
Solve for x. This will be the xcoordinate for the point of intersection.
Use this xcoordinate and substitute it into either of the original equations for the lines and solve for y. This will be the ycoordinate of the point of intersection.
To verify, substitute the xcoordinate into the other equation and you should get the same ycoordinate.
You now have the xcoordinate and ycoordinate for the point of intersection.
2. What does the intersection of two lines represent?
When the lines intersect, the point of intersection is the only point that the two graphs have in common, so the coordinates of that point are the solution for the two variables used in the equations. When the lines are parallel, there are no solutions.
3. What is the condition for intersection of two lines?
A necessary condition for two lines to intersect is that they are in the same plane i.e., they are not skew lines.
4. Can two planes intersect in a line?
They cannot intersect at only one point because planes are infinite. Furthermore, they cannot intersect over more than one line because planes are flat. One way to think about planes is to try to use sheets of paper and observe that the intersection of two sheets would only happen at one line.
5. How many solutions do same lines have?
A system of linear equations usually has a single solution, but sometimes it can have no solution (parallel lines) or infinite solutions (same line).
6. When two lines intersect how many angles are formed?
When two lines intersect, four angles are formed.
7. Do parallel lines have a solution?
Since parallel lines never cross, there can be no intersection; that is, for a system of equations that graphs as parallel lines, there can be no solution. This is called an "inconsistent" system of equations, and it has no solution.