For an LPP with constraints
\[ \begin{aligned} 2x + y &\le 70 \\ x + y &\le 40 \\ x, y &\ge 0, \end{aligned} \]
shade the feasible region on the graph and write the coordinates of its corner points.
For an LPP with constraints
\[ \begin{aligned} 2x + y &\le 70 \\ x + y &\le 40 \\ x, y &\ge 0, \end{aligned} \]
shade the feasible region on the graph and write the coordinates of its corner points.
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Solution and Explanation
Line \( 2x + y = 70 \) meets the axes at \( (35, 0) \) and \( (0, 70) \).
Line \( x + y = 40 \) meets the axes at \( (40, 0) \) and \( (0, 40) \).
The conditions \( x \ge 0 \) and \( y \ge 0 \) keep us in the first quadrant.
Step 2: Decide which side to shade:
Test the origin. \( 0 \le 70 \) and \( 0 \le 40 \) are true.
So shade the side of each line that contains the origin. The common shaded part is the feasible region.
Step 3: Find the intersection of the two lines:
Subtract \( x + y = 40 \) from \( 2x + y = 70 \). This gives \( x = 30 \), and then \( y = 10 \).
Step 4: Pick the corner points:
The shaded region is a quadrilateral. Near the x-axis the line \( 2x + y = 70 \) is tighter. Near the y-axis the line \( x + y = 40 \) is tighter.
So the corners are the origin, the x-intercept \( (35, 0) \) of the first line, the intersection point \( (30, 10) \), and the y-intercept \( (0, 40) \) of the second line.
Final Answer:
The feasible region is the shaded quadrilateral with these corner points. \[ \boxed{O(0, 0),\ A(35, 0),\ B(30, 10),\ C(0, 40)} \]
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