Answer
\[y = - \frac{5}{4}x + \frac{7}{2}\]
Work Step by Step
\[\begin{gathered}
{x^2} + xy + {y^2} = 7\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {2,1} \right) \hfill \\
\hfill \\
use\,\,the\,\,implicit\,\,differentiation \hfill \\
\hfill \\
2x + xy' + y + 2yy' = 0 \hfill \\
\end{gathered} \]
\[factor\,\,y'\]
\[y'\,\left( {x + 2y} \right) = - 2x - y\]
\[solve\,\,for\,\,y'\]
\[y' = - \frac{{2x + y}}{{x + 2y}}\]
\[\begin{gathered}
evaluate\,\,\,\,\left( {2,1} \right) \hfill \\
\hfill \\
y' = - \frac{{2\,\left( 2 \right) + 1}}{{2 + 2}} = - \frac{5}{4} \hfill \\
\hfill \\
use\,\,the\,\,point\, - \,slope\,\,form \hfill \\
\end{gathered} \]
\[\begin{gathered}
y - {y_1} = m\,\left( {x - {x_1}} \right) \hfill \\
\hfill \\
then \hfill \\
\hfill \\
y - 1 = - \frac{5}{4}\,\left( {x - 4} \right) \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
y = - \frac{5}{4}x + \frac{7}{2} \hfill \\
\end{gathered} \]
