B$.(25$ points$)$ What are Matt's ordinary demand functions, $x_1^{**}(p_1,p_2,I)$ and $x_2^{**}(p_1,p_2,I)?$

$(1)$ At corner of IC where $31=22=>x_1=\frac{2x_2}{3}$

$(2)$ and on the budget constraint, $p_1{x}_1+p_2{x}_2=I$

So$,p_1\frac{2x_2}{3}+p_2{x}_2=I=>x_2(\frac{2}{3}{p}_1+p_2)=I=>x_2^{**}(p_1,p_2,I)=\frac{\text{I}}{(\frac{2}{3}{p}_1+p_2)}$

Substitute $x^{**}$ into equation $1$ and solve for $x{1}^{**}$:$x_1=\frac{2x_2^{**}(p_1,p_2,I)}{3}=\frac{2\text{I}}{3(\frac{2}{3}{p}_1+p_2)}$