Question 63·Hard·Linear Equations in Two Variables

In the $xy$-plane, line $L$ has equation $y = \tfrac{2}{3}x + b$, where $b$ is a constant. Line $M$ is defined by $4x + 6y = 9$. The midpoint of the segment that joins the origin $(0,0)$ to the intersection point of lines $L$ and $M$ lies on the line $y = x + 1$. What is the value of $b$?