Graph the system of inequality x+2y <10,2x+y <10,x>0,y>0. Thank you
To graph the system of inequalities given by x + 2y < 10, 2x + y < 10, x > 0, and y > 0, you can follow these steps:
Step 1: Graph the first inequality, x + 2y < 10.
To graph x + 2y < 10, first, consider it as an equation (x + 2y = 10) and find its boundary line:
- Substitute x = 0, we get 2y = 10, y = 5. So, the boundary line passes through (0, 5).
- Substitute y = 0, we get x = 10. So, the boundary line also passes through (10, 0).
Now, plot the boundary line connecting the points (0, 5) and (10, 0). However, since the inequality is "less than", the line itself will not be included in the solution. Thus, draw a dashed line.
Step 2: Shade the appropriate region.
To determine which region to shade, you need to choose a point not on the line (boundary line) and substitute its coordinates into the inequality. For simplicity, let's choose (0, 0).
If we substitute x = 0 and y = 0 into the inequality x + 2y < 10:
0 + 2(0) < 10,
0 < 10.
Since the inequality holds true, shade the region below the dashed line.
Step 3: Graph the second inequality, 2x + y < 10.
To graph 2x + y < 10, again, consider it as an equation (2x + y = 10) and find its boundary line:
- Substitute x = 0, we get y = 10. So, the boundary line passes through (0, 10).
- Substitute y = 0, we get 2x = 10, x = 5. So, the boundary line also passes through (5, 0).
Plot the boundary line connecting the points (0, 10) and (5, 0) as a dashed line.
Step 4: Shade the appropriate region.
As before, choose a point not on the line (e.g., (0, 0)), substitute its coordinates into the inequality 2x + y < 10:
2(0) + 0 < 10,
0 < 10.
Since the inequality holds true, shade the region below the dashed line.
Step 5: Account for the additional inequalities, x > 0 and y > 0.
Since x > 0, you need to shade only the region to the right of the y-axis.
Similarly, since y > 0, shade only the region above the x-axis.
Step 6: Analyze the shaded region.
The shaded region that satisfies all the given inequalities will be the region where all the shaded regions from each step overlap.
In this case, the shaded region will be the triangular region below the dashed lines of both inequalities, to the right of the y-axis, and above the x-axis.
Note: It is helpful to use different colors or patterns for the shading in each step to easily differentiate the shaded regions for each inequality.
I hope this explanation helped you understand how to graph this system of inequalities.