SOLUTION: Find the minimum value of the region formed by the system of equations and functions below.
y ≥ x - 3
y ≤ 6 - 2x
2x + y ≥ - 3
f(x, y) = 3x + 4y
a. -12
b.
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-> SOLUTION: Find the minimum value of the region formed by the system of equations and functions below.
y ≥ x - 3
y ≤ 6 - 2x
2x + y ≥ - 3
f(x, y) = 3x + 4y
a. -12
b.
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Question 1143313: Find the minimum value of the region formed by the system of equations and functions below.
y ≥ x - 3
y ≤ 6 - 2x
2x + y ≥ - 3
f(x, y) = 3x + 4y
a. -12
b. -4.5
c. 9
d. 24
The objective function is 3x+4y=C where C is a constant; the objective is to find the minimum value of C under the given constraints.
Two of the constraint boundary lines have the same slope of -2: y >= -2x-3 and y <= -2x+6. The other constraint boundary line has slope 1: y >= x-3.
Make a quick sketch of the boundary lines and the resulting feasibility region.
The standard method from there is to find the corners of the feasibility region and evaluate the objective function at each corner to find the minimum value of the objective function.
There is a bit of a problem for doing that with this problem, because the feasibility region is unbounded because of the two parallel constraint boundary lines; it has only 2 corners.
But in fact you don't need to find the corners and evaluate the objective function at each corner -- either for this particular problem or any other. You can determine the correct corner by comparing the slope of the objective function to the slopes of the constraint boundary lines.
The slope of the objective function is -3/4. The minimum value of the objective function will be at the corner where a line with slope -3/4 just touches the feasibility region.
Your quick sketch should show that the corner of the feasibility region you want is (0,-3).
So the minimum value of the objective function is 3(0)+4(-3) = -12.
