is this a linear fractional transformation(LFT)? Now, suppose the transformation(in 2d) I am working with has two
separate functions
for $x$ and $y$.
That is, the transformation for $x$ is of the form
$$
x'=\frac{x}{x+y}
$$
and the transformation of $y$ is
$$
y'=\frac{y}{x+y}
$$
Each is an LFT (The schwarzian derivatives are $0$) but is
the transform as a whole still considered an LFT?
 A: I was under the impression that the term "fractional linear transformation" referred to transformations in one (generally complex) variable. However, it is certainly still valid to extend this idea to more variables; you get elements of the projective special linear groups. 
I don't think this is really the concept you're looking for, though, since your transformations have an extra property (they are homogeneous). 
A: By all conventions I know, this would not quite be even a fairly generalized "linear fractional transformation", for at least one technical-but-critical reason: the image is just the line $X+Y=1$. Possibly you are not asking the question you'd wish to ask.
But this map is "almost" a generalized linear fractional transformation: $g\in GL_3(\mathbb R)$ acts on $\mathbb R^3$ linearly, and then normalizing the third coordinate to $1$ gives the "projective" action of a sort mentioned by @QiaochuYuan:
$$
\pmatrix{a & b & c \cr d & e & f \cr g & h & i}\pmatrix{x \cr y & 1}
\;=\; \pmatrix{ax+by+c \cr dx+ey+f \cr gx+hy+i}\;\sim\; 
\pmatrix{{ax+by+c\over gx+hy+i} \cr {dx+ey+f\over gx+hy+i} \cr 1}
$$
The problem in your example is that the associated matrix would be $\pmatrix{1 & 0 & 0 \cr 0 & 1 & 0 \cr 1 & 1 & 0}$, which is not invertible.
This leads me to wonder about your larger context, and whether you'd really want to be asking a variant question, to which the answer could be a "generalized" "yes".
