Linear Equations

Introduction

This is the page on Linear Equation

Getting Started

I actually spent more time understanding how to do the matrix in wiki. An important thing to remember is you can add two equation together in a system because if you thing about 2=2, 4=4 then adding them together does not break the equality. The first time through I did not use matrices but wrote them out. Not sure which I prefer

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

In new video the put the value in a matrix so we now have \begin{bmatrix} 1 & 4 & -4 & 0 \\ 1 & 4 & 2 & 12 \\ 2 & -3 & -1 & 3 \end{bmatrix}

Multiply row 1 by -1
$R_2\to R_2+(-1).R_1$
\begin{bmatrix} -1 & 4 & -4 & 0 \\ 1 & 4 & 2 & 12 \\ 2 & -3 & -1 & 3 \end{bmatrix}

Now add the original Row 1 to Row 2 $R_1=R_1+R_2⟹(-1+1,-4+4,4+2,0+12)=(0,0,6,12)$

\begin{bmatrix} -1 & 4 & -4 & 0 \\ 0 & 0 & 6 & 12 \\ 2 & -3 & -1 & 3 \end{bmatrix}

Therefore 6z = 12

Subtract $2.R_1$ from $R_3$ $R_3\to R_3-2.R_1:$ $(2-2.1,-3-2.4,-1-2.-4,3-2.0)=(4,5,-9,3)$

\begin{bmatrix} -1 & 4 & -4 & 0 \\ 0 & 0 & 6 & 12 \\ 4 & 5 & -9 & 3 \end{bmatrix}

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  // multiply row 1 by -1
 x + 4y + 2z = 12 // No change
4x + 5y - 9z = 3  // subtract -2 .row 1 from row 3
        + 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 3rd row to top row
    -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

Infinite Solutions

Sometimes the equations are not solvable when put in echlon form. You can see this when you see maybe x = x. This means the are infinite answers and example might be

-x -y + 3z =  3
 x    + 1z =  3 
3x -y + 7z = 15

Apply Gaussian

-x -y + 3z = 3
   -y + 4z = 6
         0 = 0

Parameterizing

We can, when we have infinite solution express one or more variable in terms of the other e.g y = -6 + 4z. This is called parameterizing. There are

• leading variables
• free variables

What is a System

A system of linear equations is a collection of two or more linear equations, and a solution to a system of linear equations consists of values of each of the unknown variables in the system that satisfies all of its equations, or makes them true