juanberrio8a

2022-06-19

Find the solution set of triplets $\left(x,y,z\right)$ that fulfil this system using Gauss-Jordan:
$\left\{\begin{array}{l}-x+2z=0\\ 3x-6z=0\\ 2x-4z=0\end{array}$
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 $\left(0,0,0\right)$.
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

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
$\left(x,y,z\right)=\left(2s,t,s\right)=\left(0,0,0\right)+s\left(2,0,1\right)+t\left(0,1,0\right)$
which is a plane or 2 dimensional subspace

polivijuye

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

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