The equation $\left[\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{x}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{y}\end{array}\right]\left[\begin{array}{ccc}1& 3& 1\\ 0& 2& -1\\ 0& 0& 1\end{array}\right]\left[\begin{array}{l}1\\ \mathrm{x}\\ \mathrm{y}\end{array}\right]=\left[0\right]$ has

(i) for y = 0              (p) rational roots

(ii) for y = -1            (q) irrational roots

                               (r) integral roots

Then

$\left[\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{x}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{y}\end{array}\right]\left[\begin{array}{ccc}1& 3& 1\\ 0& 2& -1\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{l}1\\ \mathrm{x}\\ \mathrm{y}\end{array}\right]=10$

or $\left[\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3+2\mathrm{x}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1-\mathrm{x}+\mathrm{y}\end{array}\right]\left[\begin{array}{l}1\\ \mathrm{x}\\ \mathrm{y}\end{array}\right]=\left[0\right]$

or $\left[1+3\mathrm{x}+2{\mathrm{x}}^2+\mathrm{y}-\mathrm{xy}+{\mathrm{y}}^2\right]=\left[0\right]$

or $2{\mathrm{x}}^2+{\mathrm{y}}^2+\mathrm{y}+3\mathrm{x}-\mathrm{xy}+1=0$

If $\mathrm{y}=0, 2{\mathrm{x}}^2+3\mathrm{x}+1=0$

or $(2\mathrm{x}+1)(\mathrm{x}+1)=0$

or $\mathrm{x}=-1/2,-1$                (rational roots)

If $\mathrm{y}=-1, 2{\mathrm{x}}^2+4\mathrm{x}+1=0$

or $\mathrm{x}=\frac{-4\pm \sqrt{12}}{4}=\frac{-2\pm \sqrt{3}}{2}$           (irrational roots)