Question 1049706
Implicit Differentiation:

{{{2x+3(1*y+x(dy/dx))+2y*(dy/dx)=0}}}----------used quotient rule
{{{2x+3y+3x(dy/dx)+2y(dy/dx)=0}}}
{{{(3x+2y)(dy/dx)=-2x-3y}}}
{{{highlight(dy/dx=-(2x+3y)/(3x+2y))}}}



This rate of change as a slope value at (1,1) is:
{{{-(2*1+3*1)/(3*1+2*1)}}}
{{{-(5/5)}}}
{{{highlight(-1)}}}.



Assumption is that point  (1,1) is on the curve given.  The equation for line tangent to the curve at (1,1) easily found using the point-slope equation form:
{{{y-1=-1(x-1)}}}
{{{y=-x+1+1}}}
{{{highlight(y=-x+2)}}}---------in slope-intercept form