To determine the vertices of the feasible region, we will graph the system of inequalities and find the points where the lines intersect.
The first inequality, 3x + 5y ≤ 30, can be rewritten as 5y ≤ -3x + 30, which is equivalent to y ≤ (-3/5)x + 6.
The second inequality, x ≤ 9, represents a vertical line passing through x = 9.
The third inequality, y ≤ 3, represents a horizontal line passing through y = 3.
The last two inequalities, x ≥ 0 and y ≥ 0, represent the x-axis and y-axis, respectively.
We can start by graphing the last two inequalities, which form the axes:
The x-axis is the line y = 0 and the y-axis is the line x = 0.
Now let's graph the remaining inequalities:
First, plot the line y ≤ (-3/5)x + 6. To do this, plot the point (0, 6) on the y-axis and use the slope -3/5 to obtain a second point. From (0, 6), you can move 5 units to the right and 3 units down to plot a second point at (5, 3). Draw a line through these two points and shade the region below the line (including the line itself) since it represents y ≤ (-3/5)x + 6.
Next, graph the line x ≤ 9 by plotting a vertical line passing through x = 9. Shade the region to the left of the line since it represents x ≤ 9.
Finally, graph the line y ≤ 3 by plotting a horizontal line passing through y = 3. Shade the region below the line since it represents y ≤ 3.
The feasible region is the shaded region where all the shaded regions overlap. The vertices of this region are the points where the lines intersect.
Analyzing the graph, we find the following five vertices of the feasible region:
1) (0, 0)
2) (0, 3)
3) (5, 3)
4) (9, 0)
5) (9, 1.8)
These five points represent the vertices of the feasible region.
Determine the 5 vertices of the feasible region, given the constraints below. 3x+5y≤30 x≤9 y≤3 x≥0 y≥0
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1 answer