Answer
See explanations.
Work Step by Step
Step 1. Based on the given conditions, we have
$\begin{cases} 2x+3y\leq9\\x-y\leq2\\x\geq0\\y\geq0 \end{cases}$
Step 2. The objective function is $z=Ax+Bx$
Step 3. To maximize the objective function, graph the inequalities as shown and we can identify the vertices of the solution region as
$(0,0),(0,3),(3,1),(2,0)$.
Step 4. Check the value of the corner points with the objective function. We have
$z_1=A(0)+B(0)=0$; $z_2=A(0)+B(3)=3B$; $z_3=A(3)+B(1)=3A+B$; $z_4=A(2)+B(0)=2A$;
Step 5. For the same maximum value at $(3,1)$ and $(0,3)$, we have $z_2=z_3$. Thus $3B=3A+B$ which gives $A=\frac{2}{3}B$
