Let’s say you had to draw two lines on a 2D flat piece of paper, what are the possibilities for the relationship between the two lines?

1. they intersect at one point. This one should be obvious, two lines can only meet at one point. If they loop back to hit another time it is no longer a line

2. they do not intersect at one point. This is the case if you put two rulers exactly parallel from each other.

And there is technically a third option.

3. they are the same line. in which case they intersect each other infinitely many times.

Generalizing these ideas to more dimensions is a topic called linear algebra, highly recommend this series: https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

But for us we need to solve the simplest case, 2D square problems. This is detailed here: http://tutorial.math.lamar.edu/Classes/Alg/SystemsTwoVrble.aspx

Here is the problem I got:

0.5x – 0.3y = -1.1

0.3x – 0.5y = -1.3

there are many ways to do this, though I will solve for one variable and plug into the other equation:

0.5x = -1.1 + 0.3y —> x = 2(-1.1 + 0.3y) —– plug into second equation —-> 0.3(2(-1.1 + 0.3y)) – 0.5y = -1.3

this implies y = 2 after solving that linear equation for y (showing more steps than I have here). We get x now by using y = 2:

0.5x – 0.3(2) = -1.1 implies x = -1

so the one solution (the one point that the lines cross) is (-1,2) . QED

So if the two lines are parallel you would have ended up with an unsolvable/untenable equation. If the lines were the same the two equations would have been multiples of the other equation.

Discussion points

1. if you could draw the two lines in 3D instead, are there still only 3 possible configurations of two lines?

2. what are some applications of linear systems in the real world?