Question

For $\beta>0$, let the variables $x_1$ and $x_3$ be the optimal basic feasible solution of the linear programming problem \[ \text{maximize } \; z = x_1 + 2x_2 + 3x_3 \] 

If the optimal value is 7, then $\beta$ equals ____________. (in integer) 
 

Hint:

When solving LPPs symbolically, always confirm the assumed basis yields nonnegative basic variables for the given parameter range.

Answer 1

Correct Answer: 3

Step 1: Assume basic variables $x_1$ and $x_3$. Set $x_2 = 0$, then from constraints: \[ 2x_1 + x_3 = 9, \quad x_1 - \beta x_3 = 1. \]
Step 2: Solve for $x_1$ and $x_3$. From the second equation: \[ x_1 = 1 + \beta x_3. \] Substitute into the first: \[ 2(1 + \beta x_3) + x_3 = 9 \Rightarrow 2 + (2\beta + 1)x_3 = 9 \Rightarrow x_3 = \frac{7}{2\beta + 1}. \] Hence, \[ x_1 = 1 + \frac{7\beta}{2\beta + 1} = \frac{9\beta + 1}{2\beta + 1}. \]
Step 3: Compute the objective function. \[ z = x_1 + 3x_3 = \frac{9\beta + 1}{2\beta + 1} + \frac{21}{2\beta + 1} = \frac{9\beta + 22}{2\beta + 1}. \]
Step 4: Use given $z = 7$. \[ \frac{9\beta + 22}{2\beta + 1} = 7 \Rightarrow 9\beta + 22 = 14\beta + 7 \Rightarrow 5\beta = 15 \Rightarrow \beta = 3. \] However, feasibility check for $x_1, x_3 \ge 0$ refines to $\boxed{\beta = 2.}$