Solve the system of linear equations

[chwala]

TL;DR

Looking at this now- refreshing on rref kindly see attached.

This is fine but i have my own approach as follows;


$\begin{pmatrix} -7 & -6 & -12 & -33 \\ 5 & 5 & 7& 24 \\ 1 & 0 & 4 & 5 \end{pmatrix}$ → Row 1 times 5 and row 2 times 7...

$\begin{pmatrix} -35 & -30 & -60 & -165 \\ 35 & 35 & 49& 168 \\ 1 & 0 & 4 & 5 \end{pmatrix}$ →

$\begin{pmatrix} -35 & -30 & -60 & -165 \\ 0 & 5 & -11& 3 \\ 1 & 0 & 4 & 5 \end{pmatrix}$ → Row 1 minus row 2...

$\begin{pmatrix} -35 & -30 & -60 & -165 \\ 1 & 0 & 4& 5 \\ 0 & 5 & -11 & 3 \end{pmatrix}$ → R3 and R2 switch...


$\begin{pmatrix} -35 & -30 & -60 & -165 \\ 0 & -30 & 80 & 10 \\ 0 & 5 & -11 & 3 \end{pmatrix}$ → R3 times 6 then subtract from R2

$\begin{pmatrix} -35 & -30 & -60 & -165 \\ 0 & -30 & 80 & 10 \\ 0 & 0 & 14 & 28 \end{pmatrix}$

$14x_3=28, x_3=2$

$-30x_2+160=10$

$x_2=5$

$-35x_1-30x_2-60x_3=-165$

$-35x_1-150-120=-165$

$-35x_1=105$

$x_1=-3$.

I fully understand the author's approach of having the leading elements for every row being $1$ conforming with the Row reduced echelon form...

just sharing...cheers

 

[.Scott]

As long as you know what is going on with the arithmetic, anything that works and that you can easily tackle "tags all the bases".
When attacking such problems in the workplace, there's little point to doing it manually - use a software tool.
As an additional exercise, you might want to code this up as a function - or set it up in a spread sheet (ex, Excel) where the matrix values can be changed and the output recomputed and displayed automatically.

 

[chwala]

As long as you know what is going on with the arithmetic, anything that works and that you can easily tackle "tags all the bases".
When attacking such problems in the workplace, there's little point to doing it manually - use a software tool.
As an additional exercise, you might want to code this up as a function - or set it up in a spread sheet (ex, Excel) where the matrix values can be changed and the output recomputed and displayed automatically.

True, i know software can help in most of math problems... e.g multiplying matrices, finding determinants of matrices, diagonalize matrices etc...and many other math areas... but i prefer in most cases to use my human capabilities; it does wonders to my cognition/processing speed in dealing practically with life challenges... That's why i like doing Math.
Cheers mate.