Consider the region defined by the inequalities: 0≤x
0≤y
-21/4≤y-7/4x
y+x≤7
4≤y-3/7x
Pick up those points which are the boundary points of the resulting convex region.
To determine the boundary points of a convex region defined by inequalities, we need to find the points where the inequalities are equalities.
Let's go through each inequality one by one:
1) 0 ≤ x: This inequality tells us that x can take any non-negative value. So, any point on the x-axis (y = 0) will satisfy this inequality.
2) 0 ≤ y: This inequality tells us that y can take any non-negative value. So, any point on the y-axis (x = 0) will satisfy this inequality.
3) -21/4 ≤ y - 7/4x: To find the boundary points of this inequality, we need to set it equal to the inequality sign. So, y - 7/4x = -21/4. Rearranging the equation, we get y = -21/4 + 7/4x, which tells us that the boundary points lie on a line with a slope of 7/4 and y-intercept of -21/4. To find a few points on the line, we can assign some values to x and calculate the corresponding y values, or we can find the x-intercept and y-intercept.
4) y + x ≤ 7: To find the boundary points of this inequality, we need to set it equal to the inequality sign. So, y + x = 7. Rearranging the equation, we get y = 7 - x, which tells us that the boundary points lie on a line with a slope of -1 and y-intercept of 7.
5) 4 ≤ y - 3/7x: To find the boundary points of this inequality, we need to set it equal to the inequality sign. So, y - 3/7x = 4. Rearranging the equation, we get y = 4 + 3/7x, which tells us that the boundary points lie on a line with a slope of 3/7 and y-intercept of 4.
Now, we have three equations representing lines. By plotting these lines on a graph, we can find the boundary points of the resulting convex region by examining the points of intersection.