Question

The shaded region in the following figure represents the solution set for a certain linear programming problem. Then linear constraints for this region are given by}

• A. $x \ge 0, y \ge 0, x+y \le 4, x+3y \ge 6$
• B. $x \ge 0, y \ge 0, x+y \ge 4, x+3y \le 6$
• C. $x \le 0, y \le 0, x+y \le 4, x+3y \ge 6$
• D. $x \ge 0, y \ge 0, x+y \ge 4, x+3y \ge 6$

Hint:

To find linear constraints from a shaded region: (1) Identify the axes ($x \ge 0, y \ge 0$ for the first quadrant). (2) For each boundary line, find its equation (e.g., using two points or intercepts). (3) Test a point within the shaded region (or the origin if it's not in the region) to determine the correct inequality direction ($\le$ or $\ge$).

Answer 1

The Correct Option is A

Step 1: Identify the boundaries from the graph. The shaded region is in the first quadrant, which implies the basic constraints: \[ x \ge 0 \] \[ y \ge 0 \]
Step 2: Determine the equation and inequality for the first line. Observe the line that passes through the points $(0,4)$ and $(4,0)$. Using the intercept form $\frac{x}{a} + \frac{y}{b} = 1$, where $a=4$ and $b=4$: \[ \frac{x}{4} + \frac{y}{4} = 1 \] Multiply by 4: \[ x + y = 4 \] The shaded region lies below or on this line. To check, pick a point in the shaded region, e.g., $(2,2)$. $2+2=4$, which satisfies $x+y \le 4$. So, the inequality is: \[ x + y \le 4 \]
Step 3: Determine the equation and inequality for the second line. Observe the line that passes through the points $(0,2)$ and $(6,0)$. Using the intercept form $\frac{x}{a} + \frac{y}{b} = 1$, where $a=6$ and $b=2$: \[ \frac{x}{6} + \frac{y}{2} = 1 \] Multiply by the least common multiple of 6 and 2, which is 6: \[ x + 3y = 6 \] The shaded region lies above or on this line. To check, pick a point in the shaded region, e.g., $(2,2)$. $2+3(2)=8$, which satisfies $x+3y \ge 6$. So, the inequality is: \[ x + 3y \ge 6 \]
Step 4: Combine all the linear constraints. The linear constraints that define the shaded region are: \[ x \ge 0 \] \[ y \ge 0 \] \[ x + y \le 4 \] \[ x + 3y \ge 6 \]
Step 5: State the final answer. The correct set of constraints is $x \ge 0, y \ge 0, x+y \le 4, x+3y \ge 6$.