Use the chain rule to find $\frac{\partial z}{\partial s}$ and $\frac{\partial z}{\partial t}$, where $$z = x^2 + xy + y^2, x = 3 s + 7 t, y = 3 s + 9 t$$
First the pieces:
$\frac{\partial z}{\partial x} =$ $\frac{\partial z}{\partial y} =$
$\frac{\partial x}{\partial s} =$ $\frac{\partial x}{\partial t} =$
$\frac{\partial y}{\partial s} =$ $\frac{\partial y}{\partial t} =$
And putting it all together:
$\frac{\partial z}{\partial s} =$
$\frac{\partial z}{\partial t} =$