juanberrio8a

2022-06-19

Find the solution set of triplets $(x,y,z)$ that fulfil this system using Gauss-Jordan:

$\{\begin{array}{l}-x+2z=0\\ 3x-6z=0\\ 2x-4z=0\end{array}$

First of all, I don't see any $y$ variable there. I suppose it doesn't matter and I proceed normally:

$\left[\begin{array}{ccc}-1& 2& 0\\ 3& -6& 0\\ 2& -4& 0\end{array}\right]$

$-{f}_{1}$

$\left[\begin{array}{ccc}1& -2& 0\\ 3& -6& 0\\ 2& -4& 0\end{array}\right]$

$-3{f}_{1}+{f}_{2}$

$\left[\begin{array}{ccc}1& -2& 0\\ 0& 0& 0\\ 2& -4& 0\end{array}\right]$

$-2{f}_{1}+{f}_{3}$

$\left[\begin{array}{ccc}1& -2& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

So, this is the staggered reduced form.

This is an homogeneous system (because of the null column), thus one solution is $(0,0,0)$.

Other than that, I have to check out the range of the system. The range is $1$, which is less than the number of columns... what is the number of columns?

$\{\begin{array}{l}-x+2z=0\\ 3x-6z=0\\ 2x-4z=0\end{array}$

First of all, I don't see any $y$ variable there. I suppose it doesn't matter and I proceed normally:

$\left[\begin{array}{ccc}-1& 2& 0\\ 3& -6& 0\\ 2& -4& 0\end{array}\right]$

$-{f}_{1}$

$\left[\begin{array}{ccc}1& -2& 0\\ 3& -6& 0\\ 2& -4& 0\end{array}\right]$

$-3{f}_{1}+{f}_{2}$

$\left[\begin{array}{ccc}1& -2& 0\\ 0& 0& 0\\ 2& -4& 0\end{array}\right]$

$-2{f}_{1}+{f}_{3}$

$\left[\begin{array}{ccc}1& -2& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

So, this is the staggered reduced form.

This is an homogeneous system (because of the null column), thus one solution is $(0,0,0)$.

Other than that, I have to check out the range of the system. The range is $1$, which is less than the number of columns... what is the number of columns?

humusen6p

Beginner2022-06-20Added 22 answers

You are almost at the end.The number of parameters is $3-1=2$, so we have two parameter say $y=t,z=s$ and hence $x=2s$ so the set of solutions are of the form

$(x,y,z)=(2s,t,s)=(0,0,0)+s(2,0,1)+t(0,1,0)$

which is a plane or 2 dimensional subspace

$(x,y,z)=(2s,t,s)=(0,0,0)+s(2,0,1)+t(0,1,0)$

which is a plane or 2 dimensional subspace

polivijuye

Beginner2022-06-21Added 16 answers

You have two variables. and three equations. Solve for $x$ and $z$ using any two, get $0,0$. Confirm that it is consistent with your third.

Now deduce that all triples $(x,y,z)=(0,a,0)$ where $a\in \mathbb{R}$ satisfy this, because your equations don't depend on $y$.

Now deduce that all triples $(x,y,z)=(0,a,0)$ where $a\in \mathbb{R}$ satisfy this, because your equations don't depend on $y$.