A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.

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 \(a_1x + b_1y + c_1= 0\)  and \(a_2x + b_2y + c_2 = 0\), 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.

Want to find complex math solutions within seconds?
Use our free online calculator to solve challenging questions. With Cuemath, find solutions in simple and easy steps.

Book a Free Trial Class

What Is Point of Intersection Formula?

Consider 2 straight lines \(a_{1} x+b_{1} y+c_{1} = 0, \, \text{ and } \,a_{2} x+b_{2} y+c_{2} = 0\) 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:

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

Point of intersection formula

 

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

 

  1. 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,
    \(a_1 = 2, b_1 = 4, c_1 = 2\)
    \(a_2 = 2, b_2 = 3, c_2 = 5\)
    Intersection point can be calculated using the point of intersection formula,
     
    \begin{array}{l}
    \mathrm{x}=\frac{b_{1} c_{2}-b_{2} c_{1}}{a_{1} b_{2}-a_{2} b_{1}} ; \mathrm{y}=\frac{a_{2} c_{1}-a_{1} c_{2}}{a_{1} b_{2}-a_{2} b_{1}} \\
     
    (\mathrm{x}, \mathrm{y})=\left(\frac{4 \times 5-3 \times 2}{2 \times 3-2 \times 4}, \frac{2 \times 2-2 \times 5}{2 \times 3-2 \times 4}\right) \\
    (\mathrm{x}, \mathrm{y})=\left(\frac{20-6}{6-8}, \frac{4-10}{6-8}\right) \\
    (\mathrm{x}, \mathrm{y})=(-7,3)
    \end{array}

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

  2. Example 2: What will be the point of Intersection of the given two lines \(2 x+4 y+6=0\) and \(6 x+ 9y+ 12=0\).

    Solution:

    The two line equations are given as \(2 x+4 y+6=0\) and \(6x + 9y + 12=0\).
    Comparing these two equations with \(a_{1} x+b_{1} y+c_{1} and a_{2} x+b_{2} y+c_{2}\)
    We get \(a_{1}=2, b_{1}=4, c_{1}=6\)
    \(a_{2}=6, b_{2}=9, c_{2}=12\)
    Point of intersection formula is given as
    \((x, y)=\left(\frac{b_{1} c_{2}-b_{2} c_{1}}{a_{1} b_{2}-a_{2} b_{1}}, \frac{c_{1} a_{2}-c_{2} a_{1}}{a_{1} b_{2}-a_{2} b_{1}}\right)\)
    Substituting the values of \(a_{1}, b_{1}, c_{1}, a_{2}, b_{2}, c_{2}\) in the intersection formula \((x, y)=\left(\frac{4 \times 12-9 \times 6}{2 \times 9-6 \times 4}, \frac{6 \times 6-12 \times 2}{2 \times 9-6 \times 4}\right)\)
    \((x, y)=\left(\frac{48-54}{18-24}, \frac{36-24}{18-24}\right)\)
    \((x, y)=\left(\frac{-2}{-2}, \frac{4}{-2}\right)\)
    \((x, y)=(1,-2)\)

    Answer: Therefore the point of intersection of two lines \(2 x+4y+6=0 and 2x+3y+4=0 \text{is} (x, y)=(1,-2)\).

Try these >

go to slidego to slide

 

Math worksheets and
visual curriculum