In the first optimal solution of the three-variable LP, what is the value of X1?

• A. 4
• B. 5
• C. 6
• D. 0

Answer: (B)

In a linear program, the optimum often occurs at a vertex where two constraints are binding, leaving one variable as the basic variable determined by those equalities. For a three-variable problem, that means two variables are zero at the first optimal corner, and the remaining one is solved from the two tight equations.

Here, the first optimal vertex sets the other two variables to zero, so X1 is found by solving the two binding constraints. Solving those gives X1 = 5, which both satisfies feasibility and delivers the optimal objective value at that vertex. Values like 4, 6, or 0 would either violate the binding equations for that corner or correspond to a different, non-optimal vertex.