solving linear systems - with graph paper...

Let's explore a new way to solve systems of linear equations.

Assume you have something like

3x+2y=7
x+4y=9.

We draw the equations like this: 3x+2y is represented by 3 steps right, 2 up (and we mark the vector as 7, as we know that is the sum). Same thing for x+4y.

Now, we start doing jumps with the arrows. Say, doing two jumps along the "3x+2y" vector takes us to 14.

But the ultimate goal is to get to one of axes. We can do it, if we do the x+4y jump, but backward (shown as blue). As we go backward, we subtract 9.

Now, we landed on the x axis! And we ended up with 5. Where did we land? - it is 5 steps to the right yielding 5. So, each step is 1, that is x=1.

As getting to 7 takes 3 x steps (and 2 y steps), and each x step is 1, we conclude that 2 y steps are 7-3=4, and hence each y step is 2.

So the solution: x=1, y=2.

Easy.

Now, solve

4x+y=6
x+2y=5,

and

3x+4y=10
2x+2y=6.