Graph the system of inequalities:
x+3y <or= 9
3x+y <or= 9
x >or= 0
y >or= 0
To graph the system of inequalities, follow these steps:
Step 1: Graph the boundary lines for each inequality.
- For the inequality x + 3y ≤ 9, graph the line x + 3y = 9 by plotting two points on the line and drawing a line through those points. Since this inequality includes or equal to (≤), the line should be solid.
- For the inequality 3x + y ≤ 9, graph the line 3x + y = 9 by plotting two points on the line and drawing a line through those points. Again, since this inequality includes or equal to (≤), the line should be solid.
- For the inequality x ≥ 0, graph the vertical line x = 0 (y-axis) since x must be greater than or equal to 0. This line should be drawn as a dashed line because the inequality does not include or equal to (>).
- For the inequality y ≥ 0, graph the horizontal line y = 0 (x-axis) since y must be greater than or equal to 0. This line should also be drawn as a dashed line because the inequality does not include or equal to (>).
Step 2: Shade the appropriate regions.
- Shade the region that satisfies the inequality x + 3y ≤ 9. To determine which side to shade, pick a test point not on the line (0,0 is a good choice). If the test point satisfies the inequality, then shade the side where the test point is located. If not, shade the other side.
- Shade the region that satisfies the inequality 3x + y ≤ 9 (using the same methodology as above).
- Shade the region that satisfies the inequality x ≥ 0. Since this inequality is x >or= 0, shade to the right of the vertical line x = 0 (including the line itself, since it includes or equal to (>).
- Shade the region that satisfies the inequality y ≥ 0. Since this inequality is y >or= 0, shade above the horizontal line y = 0 (including the line itself).
Step 3: Determine the overlapping shaded region.
- The overlapping shaded region is the solution to the system of inequalities. It represents the values of x and y that satisfy all the given inequalities.
Note: In this case, the overlapping shaded region would be the region in the first quadrant (upper right side of the graph) that is bounded by the two solid lines x + 3y = 9 and 3x + y = 9, and also includes the x-axis and the y-axis.