Question

Why do parallel lines never meet?

Answer 1

Hint: We first have to be clear about, what are parallel lines? So, parallel lines are two lines whose perpendicular distance from any point on the line is always the line. In simpler words we can say, the lines are equidistant from each other till infinity.

Complete answer:
Two lines are said to be parallel when their slopes are equal.
Every line can be written in a generalized form as,
$y = mx + c$
Where $x,y$ are two coordinates in the axis system.
$c$ is intercept of line on y-axis (coordinate where the line intersects x-axis)
$m$ is slope of the line, it indirectly tells us inclination ($\theta $ it is the angle that the line makes from the positive x-axis) .
Value of $m$ can be determined as,
$m = \tan \left( \theta \right)$
Thus if lines are parallel both line equations will have the same slope, that is, same angle (inclination) with x-axis.
For example let's assume one equation of line $y = 6x + 3$. We can draw lines on a graph using the equation of line.
Now, assume any other line that is parallel to the above line. We can change the value of $c$ from the above equation and we can get as many parallel line equations as you need.
Let's assume $y = 6x + 5$
If we want to find if these lines intersect, the intersection point will satisfy both equations of line.
If we try to solve both equations of line for the value of $x$ and $y$ using a method of simultaneous equations, we will notice that there is no solution to these equations.
As, if we subtract these two equations, we will get,
$\left( {y + 6x} \right) - \left( {y + 6x} \right) = 5 - 3$
$ \Rightarrow 0 = 2$
Which is not possible.
This means that these two lines don't have any point in common.
Thus, parallel lines will never intersect.

Note:
We know the distance between the two lines is always constant. So, if the distance between the two lines is any non-zero positive number, then it will be constant throughout the length of the two lines. So, we can say that the distance between them will never be zero, hence, they never meet.