Subscribe to our βΆοΈ YouTube channel π΄ for the latest videos, updates, and tips.
Home | About Us | Contact Us | Privacy | Math Blog
We will learn how to find the co-ordinates of the point of intersection of two lines.
Let the equations of two intersecting straight lines be
a1 x + b1y + c1 = 0 β¦β¦β¦β¦.. (i) and
a2 x + b2 y + c2 = 0 β¦β¦.β¦... (ii)
Suppose the above equations of two intersecting lines intersect at P(x1, y1). Then (x1, y1) will satisfy both the equations (i) and (ii).
Therefore, a1x1 + b1y1 + c1 = 0 and
a2x1 + b2y1 + c2 = 0
Solving the above two equations by using the method of cross-multiplication, we get,
x1b1c2βb2c1=y1c1a2βc2a1=1a1b2βa2b1
Therefore, x1 = b1c2βb2c1a1b2βa2b1 and
y1 = c1a2βc2a1a1b2βa2b1, a1b2 - a2b1 β 0
Therefore, the required co-ordinates of the point of intersection of the lines (i) and (ii) are
(b1c2βb2c1a1b2βa2b1, (c1a2βc2a1a1b2βa2b1), a1b2 - a2b1 β 0
Notes: To find the coordinates of the point of intersection of two non-parallel lines, we solve the given equations simultaneously and the values of x and y so obtained determine the coordinates of the point of intersection.
If a1b2 - a2b1 = 0 then a1b2 = a2b1
β a1b1 = a2b2
β - a1b1 = - a2b2 i.e., the slope of line (i) = the slope of line (ii)
Therefore, in this case the straight lines (i) and (ii) are parallel and hence they do not intersect at any real point.
Solved example to find the co-ordinates of the point of intersection of two given intersecting straight lines:
Find the coordinates of the point of intersection of the lines 2x - y + 3 = 0 and x + 2y - 4 = 0.
Solution:
We know that the co-ordinates of the point of intersection of the lines a1 x+ b1y+ c1 = 0 and a2 x + b2 y + c2 = 0 are
(b1c2βb2c1a1b2βa2b1, (c1a2βc2a1a1b2βa2b1), a1b2 - a2b1 β 0
Given equations are
2x - y + 3 = 0 β¦β¦β¦β¦β¦β¦β¦β¦.. (i)
x + 2y - 4 = 0 β¦β¦β¦β¦β¦β¦β¦β¦.. (ii)
Here a1 = 2, b1 = -1, c1 = 3, a2 = 1, b2 = 2 and c2 = -4.
((β1)β (β4)β(2)β (3)(2)β (2)β(1)β (β1), (3)β (1)β(β4)β (2)(2)β (2)β(1)β (β1))
β (4β64+1, 3+84+1)
β (115,β25)
Therefore, the co-ordinates of the point of intersection of the lines 2x - y + 3 = 0 and x + 2y - 4 = 0 are (115,β25).
β The Straight Line
11 and 12 Grade Math
From Point of Intersection of Two Lines to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Jul 29, 25 12:59 AM
Jul 28, 25 01:52 PM
Jul 25, 25 03:15 AM
Jul 24, 25 03:46 PM
Jul 23, 25 11:37 AM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.