Alright, let's solve this system of equations whimsically, shall we?
First, imagine these lines as threads of destiny. Each equation is like a thread weaving through the vast fabric of the cosmos, plotted on the 2D plane of our imagination.
So let's embark on this mathematical journey together!
Our first traveler is Line 1, or as we'll call him, "Sir L1." He struts along with the equation y = 3x + 1. His path is clear: for every step he takes in the x direction, he ascends 3 steps in the y. He's quite the climber!
Next, we meet Lady L2 with the equation y = 2x - 3. A bit more relaxed than Sir L1, she ascends 2 steps in the y direction for every step in the x. They clearly don't cross paths—they're too different in their inclinations. This means that Sir L1 and Lady L2 are consistent and independent.
Our third traveler is the mysterious Master L3. His equation is slightly different, written as 3y + x = 6. Let's write this in y = mx + c form to match our other travelers. With a little rearranging, we get y = -1/3x + 2. He's a bit of a downer, descending 1 step in y for every 3 steps in x. Observing him from afar, it's clear he never intersects with Lady L2 or Sir L1. They're all leading their independent lives.
Lastly, we have the young Miss L4, with her equation y = 1/3x - 1. She's a gentle traveler, ascending only a third of a step in y for every step in x. However, like the others, she doesn't cross paths with any of our other travelers.
So, in this whimsical world of equations, it appears that none of our travelers cross paths. Each one treads their own unique path. This means our system of equations is indeed consistent, but each line is independent, meaning they don't intersect, and each has a unique solution. There are no shared points of intersection. It seems our travelers are lone wolves in this cosmos of coordinates!