Linear Equations

Introduction

This is the page on Linear Equation

Getting Started

Already fine with this but did pick up a new technique. In the video we had the following to solve.

 x + 4y -4z = 0
 x + 4y + 2z = 12
 2x -3y -z = 3

He said for the middle row by times the top row by -1 and add it to the middle to transform. I struggled to understand as my approach is to move the values around to get to one e.g for the top one I would make it

x = 4z -4y

And replace all the instances of x but his approach is to make the x amount 0 by multiplying by whatever would make it 0 when added to the next row. So

-x - 4y + 4z = 0  // top row multiplied by -1
 x + 4y + 2z = 12 // now add middle row to it
        + 6z = 12 // Removed the x

Need for me some explanation given I have never seen this approach. For the 3rd line we multiply by -2 and add to row 3

-2x - 8y + 8z = 0 // top row multiplied by -2
 2x  -3y - 1z = 3 // now add third row to it
    -11y + 7z = 3 // Removed the x

Now we have

 x + 4y - 4z = 0
   -11y + 7z = 3 
        + 6z = 12

This is called Gaussian elimination method. Where you are left with 3 equations with 3, 2 and one variables and is known as the echelon form