Examples of Matrix Elimination to Solve a System

$$\{\begin{array}{l}3x+2y=10\\ 6x+4y=c\end{array}$$ $$⟺\left[\begin{array}{cccc}3& 2& |& 10\\ 6& 4& |& c\end{array}\right]$$ $$⟺\left[\begin{array}{cccc}3& 2& |& 10\\ 0& 0& |& c-20\end{array}\right]$$ $$⟺\{\begin{array}{ll}c=20& :\mathbb{S}=\{\left[\begin{array}{c}x\\ 5-\frac{3}{2}x\end{array}\right]\in {\mathbb{R}}^2:x\in \mathbb{R}\}\\ c\ne 20& :\mathbb{S}=\varnothing \end{array}$$

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Determine whether $\mathbf{u}\in \text{span}(\mathbf{v},\mathbf{w})$, where…

$$\mathbf{u}=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right],\mathbf{v}=\left[\begin{array}{c}1\\ 0\\ 3\end{array}\right],\mathbf{w}=\left[\begin{array}{c}1\\ 1\\ 3\end{array}\right]$$

$\mathbf{u}\in \text{span}(\mathbf{v},\mathbf{w})⟺$ there are scalars $x,y$ such that $x\mathbf{v}+y\mathbf{w}=\mathbf{u}$,

$$\left[\begin{array}{cccc}1& 1& |& 1\\ 0& 1& |& 2\\ 3& 3& |& 3\end{array}\right]$$ $$⟺\left[\begin{array}{cccc}1& 0& |& 3\\ 0& 1& |& 2\\ 0& 0& |& 0\end{array}\right]$$ $$⟺x=3,y=2\therefore \text{Yes}$$