Let $A(2,-3)$ and $B(-2,1)$ be vertices of a triangle $ABC.$ If the centroid of this triangle moves on the line $2x+3y=1,$ then the locus of the vertex $C$ is the line

Let the coordinates of vertex $C$ be $(h,k)$.
The vertices are $A(2,-3),B(-2,1),$ and $C(h,k).$
The centroid $G$ of triangle $ABC$ is:
$G=(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})=(\frac{2-2+h}{3},\frac{-3+1+k}{3})=(\frac{h}{3},\frac{k-2}{3})$.
It is given that the centroid moves on the line $2x+3y=1$.
Substitute the coordinates of $G$ into the line equation:
$2(\frac{h}{3})+3(\frac{k-2}{3})=1$.
$\frac{2h}{3}+(k-2)=1$
Multiply by $3$:
$2h+3(k-2)=3$
$2h+3k-6=3$
$2h+3k=9$
Replacing $(h,k)$ with $(x,y)$ gives the locus:
$2x+3y=9$.