Your system can be easily solved explicitely so the function can be written as:
function [s,t] = intersectionPoint2(a,b,c,d)
aa = (b - a);
bb = (c - d);
cc = (c - a);
det = (-aa(2)* bb(1) + aa(1)* bb(2));
s = (bb(2)* cc(1) - bb(1) *cc(2))/det;
t = -(aa(2)* cc(1) - aa(1) *cc(2))/det;
endfunction
which is (very slightly) faster:
tic, for ii=1:1e5, [s,t] = intersectionPoint2(a,b,c,d); endfor; toc
Elapsed time is 13.16 seconds.
tic, for ii=1:1e5, [s,t] = intersectionPoint(a,b,c,d); endfor; toc
Elapsed time is 21.2 seconds.