In this publication, we will consider what is the point of intersection of two lines, and how to find its coordinates in different ways. We will also analyze an example of solving a problem on this topic.
Finding the coordinates of the point of intersection
intersecting Lines that have one common point are called.
M is the point of intersection of the lines. It belongs to both of them, which means that its coordinates must simultaneously satisfy both of their equations.
To find the coordinates of this point on the plane, you can use two methods:
- graphic – draw graphs of straight lines on the coordinate plane and find their intersection point (not always applicable);
- analytical is a more general method. We combine the equations of lines into a system. Then we solve it and get the required coordinates. How the lines behave with respect to each other depends on the number of solutions:
- one solution – intersect;
- the set of solutions are the same;
- no solutions – parallel, i.e. do not intersect.
Example of a problem
Find the coordinates of the point of intersection of the lines
Solution
Let’s make a system of equations and solve it:
In the first equation, we express x via y:
x = y – 6
Now we substitute the resulting expression into the second equation instead of x:
y = 2 (y – 6) – 8
y = 2y – 12 – 8
y – 2y = -12 – 8
-y = -20
y = 20
Hence, x = 20 – 6 = 14
Thus, the common point of intersection of the given lines has coordinates