# How does a system of linear equations have no solution?

May 7, 2018

I tried this:

#### Explanation:

To visualize this situation we can use a simple example:
Consider two linear equations in $x \mathmr{and} y$ representing a straight line each.

Solving a system involving these two equations can lead us to find, for example, one solution...but what does this means?
We find a set of coordinates ${x}_{0} \mathmr{and} {y}_{0}$ that substituted into our equations satisfy both. In geometrical terms we find a point in common between the two straight lines or the point where the two lines cross each other.

It can happen that the two lines do not cross....the lines are parallel. In this case we cannot find a point in common between the two and consequently the system of the two equations representing the two lines will not give us a solution!

Example:
consider the two equations:

$y = 3 x + 5$
$y = 3 x - 3$

if you try to solve the system of these two equations (by substitution, for example) you'll get a strange situation....no solutions!
If you plot the two lines corresponding to the two equations you will see that they are parallel!