Graph solution region of the following system of linear inequalities by shading.10𝑥 + 20𝑦 ≤ 140 ; 6𝑥 + 18𝑦 ≥ 72 ; 𝑥 ≥ 0 ; 𝑦 ≥ 0

To graph the solution region of the system of linear inequalities, we will follow these steps:

Step 1: Graph the boundary lines of each inequality.

For the first inequality, 10𝑥 + 20𝑦 ≤ 140, we can rewrite it as 𝑦 ≤ -0.5𝑥 + 7. To graph this line, we start by plotting the y-intercept at (0, 7), and then use the slope of -0.5 to find additional points. We can choose another point by setting 𝑥 = 2, which gives us 𝑦 = 6. Connect these points to draw the line.

For the second inequality, 6𝑥 + 18𝑦 ≥ 72, we can simplify it to 𝑦 ≥ -0.33𝑥 + 4. To graph this line, we start by plotting the y-intercept at (0, 4), and then use the slope of -0.33 to find additional points. We can choose another point by setting 𝑥 = 3, which gives us 𝑦 = 3. Connect these points to draw the line.

Step 2: Determine the region of the graph that satisfies all the inequalities.

To determine the shaded region that satisfies all the inequalities, we need to consider the overlapping region between the two lines we just graphed. This region represents the solution to the system of inequalities.

Step 3: Shade the solution region.

Shade the region below the line 𝑦 ≤ -0.5𝑥 + 7 and above the line 𝑦 ≥ -0.33𝑥 + 4. Also, shade the region to the right of the y-axis (𝑥 ≥ 0) and above the x-axis (𝑦 ≥ 0). The shaded region represents the solution to the system of inequalities.

Step 4: Label the axes and any important points.

Label the x-axis and y-axis accordingly. You can also label the points where the lines intersect, if applicable.

That's it! You have now graphed the solution region of the given system of linear inequalities by shading.