Your surface is defined by a polynomial equation $f(x,y,z) = 0$. You could interpret this situation geometrically in the following way:
$f:R^3\to R$ gives a color to every point of the 3D space (imagine you have a color palette indexed by real numbers). Assume white is indexed at $0$, then your surface is the locus of points with white color.
The gradient $$ \nabla f = ( \partial_x f, \partial_y f, \partial_z f ) $$
is orthogonal to the surface when evaluated at points $(x,y,z)$ of the surface. This is because the color stays constant in our white surface.
So it's enough for you to compute $\nabla f$ with partial derivatives, evaluate it at your point $(-1,1,-3)$ to get a vector $V$, and then notice that the set of vectors perpendicular to that one is given by the equation
$$ V\cdot (X,Y,Z) = 0 $$
Doing this you find the equation of a plane parallel to the tangent space, but passing through the origin. To get the tangent space, translate it via $(-1,1,-3)$. Finally, you should get
$$ Z=15X+9Y+3 $$