If two coplanar lines are not parallel then they intersect each other. Furthermore, if they are placed on the Cartesian plane, we can determine the coordinates of the point of intersection.
Suppose we are given the line g and h whose equations are A1x + B1y + C1 = 0 and A2x + B2y + C2 = 0, respectively. Assume that the lines are not parallel. Let the point of intersection be P(x*,y*). (See the Figure 1.)
Figure 1
Since P lies on both lines, it follows that A1x* + B1y* + C1 = 0 and A2x* + B2y* + C2 = 0. As g is not parallel to h, A1B2 – A2B1 ≠ 0 and consequently the values of x* and y* can be determined by various methods to solve a system of linear equations with two unknowns. All the methods will result in:
Example 1
Given the lines having the equations y = x + 4 and y = 2x + 1, find the coordinates of the point of intersection and draw the lines on the Cartesian plane.
Answer
Applying the method of substitution, replace y in the equation y = 2x + 1 with x + 4, we get the equation x + 4 = 2x + 1. Note that:
x + 4 = 2x + 1 ⇔ x – 2x = 1 – 4 ⇔ – x = – 3 ⇔ x = 3
Substitute x = 3 into y = x + 4 or y = 2x + 1, then we have y = 7. So, the lines intersect at (3,7).
Figure 2
Example 2
Given the line g and h whose equations are 2x + 3y + 1 = 0 and 5x – 3y – 8 = 0, respectively, find the coordinates of the point of intersection by applying the method of elimination and draw the lines on the Cartesian plane.
Answer
g: 2x + 3y + 1 = 0 ……………………… (1)
h: 5x – 3y – 8 = 0 ……………………… (2)
To find x, eliminate y by adding the equation (1) to (2):
2x + 3y + 1 = 0
5x – 3y – 8 = 0 +
7x – 7 = 0
7x = 7
x = 1
To find y, eliminate x by multiplying (1) by 5 and multiplying (2) by (-2), and then add the resulting equations.
5.(1) ……………………… 10x + 15y + 5 = 0
-2.(2) …………………….. -10x + 6y + 16 = 0 +
21y + 21 = 0
21y = -21
y = -1
So, the lines intersect at (1,-1).
Applying the formulae at the beginning of this post to find x* and y*, Example 2 can also be solved as follows.
From the equations 2x + 3y + 1 = 0 and 5x – 3y – 8 = 0 we have A1 = 2, B1 = 3, C1 = 1, A2 = 5, B2 = -3, C2 = -8.
Substituting these values into the formulae, it follows that:
So, the lines intersect at (1,-1).
Figure 3
As a matter of fact, there are other methods to find the coordinates of the point of intersection of two lines. In essence, finding the coordinates of the point of intersection of any two lines whose equations are known is performed by finding solutions/roots of the system of linear equations corresponding to the equations of the lines.
