Question 309003
Complete the square in x and y. 
{{{x^2+3y^2-16x+24y+31=0}}}
{{{x^2-16x+3y^2+24y+31=0}}}
{{{(x^2-16x+64)+3(y^2+8y+16)+31=64+48}}}
{{{(x-8)^2+3(y+4)^2+31=112}}}
{{{(x-8)^2+3(y+4)^2=81}}}
{{{(x-8)^2/81+(y+4)^2/27=1}}}
Unequal coefficients, equal signs points to an ellipse.

{{{ graph( 300, 300, -5, 20, -12, 12, (-4+sqrt(81-(x-8)^2))/3, (-4-sqrt(81-(x-8)^2))/3) }}}