Intersection of Two Lines

Two distinct lines intersect at the most at one point. To find the intersection of two lines we just need to solve their equations. The alternative way is to graph the lines and find their point of intersection.

The lines will intersect only if they are non-parallel lines. Common examples of real-life intersecting lines include a pair of scissors, a folding chair, a road cross, a signboard, etc. In this mini-lesson, we will learn in detail, how to find the point of intersection of two lines.

Intersection of two lines is a point at which both lines meet. When two lines share a common point, they are called intersecting lines. This common point that exists on all intersecting lines is called the point of intersection. The two non-parallel straight lines which are co-planar will have an intersection point. Here, lines A and B intersect at point O, which is the "point of intersection".

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. 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 Cross multiplication 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

\(\left( {{x_0},{y_0}} \right) = \left( {\frac{{{b_1}{c_2} - {b_2}{c_1}}}{{{a_1}{b_2} - {a_2}{b_1}}},\frac{{{c_1}{a_2} - {c_2}{a_1}}}{{{a_1}{b_2} - {a_2}{b_1}}}} \right)\)

☛Also Check: Check the following alternative methods to find point of intersection of lines:

• Substitution Method
• Elimination Method
• Cramer's rule

To obtain the angle of intersection of lines, consider the figure shown:

The equations of the two lines in slope-intercept 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 \(\theta = {\theta _2} - {\theta _1}\), where \(\theta _1\) and \(\theta _2\) are the angles made by the two lines with the x-axis. 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.

• 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, θ = π/2 , so that cot θ = 0; this can happen if \(1 + {m_1}{m_2} = 0\) or \({m_1}{m_2} = - 1\).

We can another set of conditions using the above. For that, consider the following formula to find the slope of a line from its equation:

The slope of a line ax + by + c = 0 in general is m = -a/b. Now, consider two lines \(L_1: a_1x+b_1y+c_1=0\) and \(L_2: a_2x+b_2y+c_2=0\).

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 = 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 \(1(1)+1(-1)=0\).

• 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^∘ and less than 180^∘.
• 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,

• ∠a & ∠c and ∠b & ∠d are pairs of vertical angles.
• ∠a + ∠d = ∠b + ∠c = straight angle =180^∘
• An acute angle \(\theta \) between lines \(L_1\) and \(L_2\) with slopes \(m_1\) and \(m_2\) is given by \(\theta = {\tan ^{ - 1}}\left| {\frac{{{m_2} - {m_1}}}{{1 + {m_1}{m_2}}}} \right|\).
• 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: \({m_1}{m_2} = - 1\)

☛Related articles:

• System of Equations Calculator
• Simultaneous Equations Calculator
• Point of Intersection of Lines Calculator

How to Find Intersection of Two Lines?

To find the point of intersection of two lines:

• Get the two equations for the lines into slope-intercept 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 x-coordinate for the point of intersection.
• Use this x-coordinate and substitute it into either of the original equations for the lines and solve for y. This will be the y-coordinate of the point of intersection.
• You now have the x-coordinate and y-coordinate for the point of intersection.

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. i.e., the lines don't intersect.

What is the Condition for the 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.

What are the Methods Used to Find Point of Intersection of Lines?

The following are different methods used to find the point of intersection of two lines:

• Substitution method
• Elimination method
• Graphical method
• Cross multiplication method
• Cramer's rule
• Matrix method

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 along one line.

How Many Solutions Do the 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).

When Two Lines Intersect How Many Angles are Formed?

When two lines intersect, four angles are formed. Among them, two are acute and two are obtuse.

Do Parallel Lines Have a Solution?

Since parallel lines never cross, there can be no intersection; 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.