Have i solved the question correctly?

Dear Student,

Yes,your first solution is correct,
and second solution is incomplete

$$\begin{gathered} 5\frac{1}{4}÷2\frac{1}{3}-4\frac{2}{3}+5\frac{1}{3}\times 3\frac{1}{2} \\ =\frac{21}{4}÷\frac{7}{3}-\frac{14}{3}+\frac{16}{3}\times \frac{7}{2} \\ =\frac{21}{4}\times \frac{3}{7}-\frac{14}{3}+\frac{16}{3}\times \frac{7}{2} \\ =\frac{\cancel{{21}^3}}{4}\times \frac{3}{\cancel{7}}-\frac{14}{3}+\frac{\cancel{{16}^8}}{3}\times \frac{7}{\cancel{2}} \\ =\frac{9}{4}-\frac{14}{3}+\frac{56}{3} \\ =\frac{9}{4}-\frac{14+56}{3} \\ =\frac{9}{4}-\frac{70}{3} \\ =\frac{27-280}{12} \\ =\frac{-253}{12} \\ =\mathbf{-}\mathbf{21}\frac{\mathbf{1}}{\mathbf{12}} \end{gathered}$$
Regards