By graphing the system of constraints, find the values of x and y that minimize the objective function.

Question

x+2y>=8
x>=2
y>=0
minimum for C = x + 3y

Answer 1

To graph the system of constraints, we'll start with the inequalities:

1) x + 2y ≥ 8
2) x ≥ 2
3) y ≥ 0

First, let's graph the line for equality in the first constraint: x + 2y = 8.
To find the x and y-intercepts, let's set x = 0:
0 + 2y = 8
2y = 8
y = 4
So, one point on the line is (0, 4).
Now, let's set y = 0:
x + 2(0) = 8
x = 8
The second point on the line is (8, 0).
Drawing a line through these two points, we have the boundary line for the first constraint.

Next, let's graph the equality for the second constraint: x = 2.
This is a vertical line passing through (2,0).

Lastly, for the third constraint: y ≥ 0, this is the positive y-axis.

Graphing these lines and regions, we find:

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| . O
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|_____|

The shaded region represents the feasible region where all the constraints are satisfied.

Now, let's graph the objective function: C = x + 3y.
We can plot some points and connect them to make a line going through the feasible region as follows:

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4 | . O
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3 | . | .
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|_____|
2 4

We can see that the line representing "C = x + 3y" is below the shaded region. The minimum value of the objective function occurs at the vertex in the shaded region closest to the line.

By analyzing, the vertex nearest to the line "C = x + 3y" is (2, 2).

Therefore, the values of x and y that minimize the objective function C = x + 3y are x = 2 and y = 2.