Thanks @Chestermiller! I am able to get the correct answer using this method. However, I am not exactly sure why the reasoning of finding the point up until the system receives heat does not work.
@TSny you're right, I meant the adiabatic process being tangent to the linear process.
Another way...
A linear transformation is described by:
$$p=a-bV$$
From that we can find $a$ and $b$:
$$b=\frac{p_{2}-p_{1}}{V_{1}-V_{2}} = \frac{2}{9}\frac{p_{1}}{V_{1}}$$
$$a=p_{1}+bV_{1}=\frac{11}{9}p_{1}$$
I considered an adiabatic process that intersects the linear transformation to find the point up...