Suppose that y≤5x,3x≤y and 14x+15y≤1 together with 0≤x, 0≤y.
The maximum value of the function y+x on the resulting region occurs at x=? and y=?
The maximum value of the function is ?
To find the maximum value of the function y+x on the given region, we need to determine the values of x and y that satisfy the given inequalities and maximize the value of y+x.
Let's analyze the given inequalities one by one:
1. y ≤ 5x
This inequality suggests that y is less than or equal to 5 times x. It represents a region below the line y = 5x on the coordinate plane.
2. 3x ≤ y
This inequality indicates that 3 times x is less than or equal to y. It represents a region above the line y = 3x on the coordinate plane.
3. 14x + 15y ≤ 1
This inequality represents a region below the line 14x + 15y = 1 on the coordinate plane.
The region that satisfies all three inequalities is the intersection of the regions defined by each inequality.
Now, let's find the intersection points of the lines y = 5x, y = 3x, and 14x + 15y = 1:
1. Intersection of y = 5x and y = 3x:
Setting the two equations equal to each other, we get:
5x = 3x
2x = 0
x = 0
Substituting this value of x into either equation, we find y = 0.
So, the intersection point of y = 5x and y = 3x is (0, 0).
2. Intersection of y = 3x and 14x + 15y = 1:
Substituting y = 3x into 14x + 15y = 1, we get:
14x + 15(3x) = 1
14x + 45x = 1
59x = 1
x = 1/59
Substituting this value of x into y = 3x, we find y = 3(1/59) = 3/59.
So, the intersection point of y = 3x and 14x + 15y = 1 is (1/59, 3/59).
Now, let's check the values of x and y in the region defined by the inequalities:
1. For x = 0 and y = 0, we have:
y + x = 0 + 0 = 0
2. For x = 1/59 and y = 3/59, we have:
y + x = (3/59) + (1/59) = 4/59
Comparing these values, we can see that the maximum value of y + x is 4/59, which occurs at x = 1/59 and y = 3/59.
So, the maximum value of the function y + x on the resulting region is 4/59.