Question

Minimize $z=x+y$ subject to $2x+3y \ge 6, x \ge 0, y \ge 0$. The solution is ________.

• A. 0
• B. 2
• C. 3
• D. 5
• E. 6

Hint:

Test all corner points in the objective function.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
This is a linear programming problem (LPP). We need to find the minimum value of an objective function \(z = x + y\) subject to a set of linear constraints. The solution to an LPP, if it exists, will occur at one of the corner points (vertices) of the feasible region defined by the constraints.
Step 2: Key Formula or Approach
1. Graph the feasible region. To do this, treat the inequalities as equations to find the boundary lines, and then use the inequality sign to determine which side of the line is included in the region.
2. Identify the corner points of the feasible region.
3. Evaluate the objective function \(z = x + y\) at each corner point.
4. The smallest value of \(z\) obtained will be the minimum value.
Step 3: Detailed Explanation
1. Graph the feasible region.
- The constraints are \(x \ge 0\) and \(y \ge 0\). This means the feasible region is in the first quadrant.
- The main constraint is \(2x + 3y \ge 6\). Let's first plot the boundary line \(2x + 3y = 6\). - To find the x-intercept, set \(y=0\): \(2x = 6 \implies x = 3\). The point is (3, 0). - To find the y-intercept, set \(x=0\): \(3y = 6 \implies y = 2\). The point is (0, 2). - Draw a line connecting (3, 0) and (0, 2). - The inequality is \(2x + 3y \ge 6\). To determine the region, we can test the origin (0, 0): \(2(0) + 3(0) = 0\), which is not \(\ge 6\). So, the feasible region is the area on the side of the line that does not contain the origin (i.e., the area above the line).
- The feasible region is unbounded, extending infinitely in the first quadrant above the line \(2x+3y=6\).
2. Identify the corner points.
The feasible region is an unbounded area in the first quadrant. The corner points of this region are the intercepts of the line \(2x + 3y = 6\) with the axes.
The corner points are:
- A = (3, 0)
- B = (0, 2)
3. Evaluate the objective function at the corner points.
The objective function is \(z = x + y\).
- At point A(3, 0): \(z = 3 + 0 = 3\).
- At point B(0, 2): \(z = 0 + 2 = 2\).
4. Determine the minimum value.
Comparing the values of \(z\) at the corner points, the smallest value is 2.
Since the feasible region is unbounded, we must check if an even smaller value is possible. We check the inequality \(z<2\), which is \(x+y<2\). The region defined by \(x+y<2\) is a half-plane below the line \(x+y=2\). This region has no points in common with the feasible region. Therefore, the minimum value must occur at a corner point.
The minimum value of \(z\) is 2.
Step 4: Final Answer
The minimum value of z is 2.