$\%$ Define the functions representing the equations

f$1=$ @$($x$,$ y$)-$x$^2+$ x $+0\mathrm{.}75-$ y;

f$2=$ @$($x$,$ y$)$ y $+5*$x$*$y $-$ x$^2$;

$\%$ Define the partial derivatives of the functions with respect to x and y

df$1\_$dx $=$ @$($x$,$ y$)-2*$x $+1$;

df$1\_$dy $=$ @$($x$,$ y$)-1$;

df$2\_$dx $=$ @$($x$,$ y$)-2*$x $+5*$y;

df$2\_$dy $=$ @$($x$,$ y$)1+5*$x;

$\%$ Initial guesses

x$0=1\mathrm{.}2$;

y$0=1\mathrm{.}2$;

$\%$ Set tolerance and maximum number of iterations

tolerance $=1$e$-6$;

maxIterations $=100$;

$\%$ Initialize variables

x $=$ x$0$;

y $=$ y$0$;

iteration $=0$;

while iteration $<$ maxIterations

$\%$ Evaluate the functions and their derivatives at the current point

f$1\_$val $=$ f$1($x$,$ y$)$;

f$2\_$val $=$ f$2($x$,$ y$)$;

J $=[$df$1\_$dx$($x$,$ y$),$ df$1\_$dy$($x$,$ y$)$; df$2\_$dx$($x$,$ y$),$ df$2\_$dy$($x$,$ y$)]$;

$\%$ Calculate the update using the Newton$-$Raphson formula

update $=-$J $\backslash [$f$1\_$val; f$2\_$val$]$;

$\%$ Update x and y

x $=$ x $+$ update$(1)$;

y $=$ y $+$ update$(2)$;

$\%$ Check for convergence

if norm$($update$)<$ tolerance

fprintf$(\text{'}$Converged after $\%$d iterations.

$\text{'},$ iteration$)$;

fprintf$(\text{'}$Solution: x $=\%\mathrm{.}6$f$,$ y $=\%\mathrm{.}6$f

$\text{'},$ x$,$ y$)$;

break;

end

iteration $=$ iteration $+1$;

end

if iteration $>=$ maxIterations

fprintf$(\text{'}$Newton$-$Raphson did not converge after $\%$d iterations.

$\text{'},$ maxIterations$)$;

end