Answer: To graph the system of inequalities, let's first graph the boundary lines for each inequality:
- Z = x + 3y (This is a plane, not an inequality)
- y ≥ -2x + 2 (This is a line with a shaded region above it)
- y ≤ 3x + 2 (This is a line with a shaded region below it)
- y ≤ -5x + 15 (This is a line with a shaded region below it)
Step 1: Graph the lines:
To graph each line, first, find two points that satisfy the equation and then draw a straight line passing through those points.
Equation 2 (y ≥ -2x + 2):
For x = 0, y = 2 (point: (0, 2))
For x = 1, y = 0 (point: (1, 0))
Plot these points and draw a line passing through them. Since y is greater than or equal to the line, shade the region above the line.
Equation 3 (y ≤ 3x + 2):
For x = 0, y = 2 (point: (0, 2))
For x = 1, y = 5 (point: (1, 5))
Plot these points and draw a line passing through them. Since y is less than or equal to the line, shade the region below the line.
Equation 4 (y ≤ -5x + 15):
For x = 0, y = 15 (point: (0, 15))
For x = 1, y = 10 (point: (1, 10))
Plot these points and draw a line passing through them. Since y is less than or equal to the line, shade the region below the line.
Step 2: Identify the feasible region:
The feasible region is the shaded region where all three inequalities overlap.
Step 3: Plot Z = x + 3y:
Since Z = x + 3y represents a plane, we can plot some points to help visualize it:
For x = 0, y = 0, Z = 0 (point: (0, 0, 0))
For x = 1, y = 1, Z = 4 (point: (1, 1, 4))
Plot these points and draw the plane passing through them.
The graph will show the feasible region where all the inequalities are satisfied, along with the plane Z = x + 3y.