Suppose that, we have a homothetic transformation in a rectangular coordinate system, with center origin $(0,0)$ and $k$. This homothetic sends point $A(2,3)$ to point $B(2x-1,x)$. My aim is to find $k$. First I have sketched the equation of line that goes through the origin and point $(2,3)$ and it has the form:
$$y=3x/2.$$
Now I know that a homothetic does it (http://en.wikipedia.org/wiki/Homothetic_transformation) which means that the point $(2x-1,x)$ is on $y=3x/2$ and the line which goes through point $(2x-1,x)$ represented as $y=3x/2$ multiplied by $k$.
How can i solve the equation with both $x$ and $k$? Or should I simply take some value of x and solve it?
If $O(0,0)$ is the center of homothety and $k$ is its ratio, the point $A$ is sent to point $B$ according to the following equation: $$B= O+k(A-O)$$ Substituting $A$ and $B$, we get: $$(2x-1,x)=(0,0)+k(2,3)$$ Hence $x=3k$ and $2x-1=2k$.
Solving that system of equations, we get $k=\frac{1}{4}$ and $x=\frac{3}{4}$.