To find the minimum value of [tex]\( C = 2x + y \)[/tex] subject to the given constraints, we must identify the feasible region defined by these constraints and determine where the objective function [tex]\( C \)[/tex] attains its minimum value.
1. List the constraints:
[tex]\[
\begin{cases}
2x + y \leq 10 \\
x - 3y \geq -3 \implies -x + 3y \leq 3 \\
x \geq 0 \\
y \geq 2
\end{cases}
\][/tex]
2. Determine the intersection points of these constraints within the feasible region:
- To find where [tex]\( 2x + y \leq 10 \)[/tex] intersects [tex]\( y = 2 \)[/tex]:
[tex]\[
2x + 2 \leq 10 \implies 2x \leq 8 \implies x \leq 4 \quad \text{(Intersection point: } (4, 2) \text{)}
\][/tex]
- To find where [tex]\( -x + 3y \leq 3 \)[/tex] intersects [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 2 \)[/tex]:
[tex]\[
y = 2 \implies -x + 3(2) \leq 3 \implies -x + 6 \leq 3 \implies -x \leq -3 \implies x \geq 3 \quad \text{(Intersection point: } (3, 2) \text{)}
\][/tex]
Note that check for validity confirms that (3, 2) satisfies [tex]\(2x + y \leq 10\)[/tex].
- Checking boundary conditions:
[tex]\[
\text{For } y = 2, \text{ constraints } 2x + y \leq 10 \text{ reduces to } 2x + 2 \leq 10 \implies 2x \leq 8 \implies x \leq 4.
\][/tex]
3. Identify the feasible points within the region that satisfy all constraints:
By identifying the intersection points and boundary constraints, we find that the primary feasible points of interest are:
- [tex]\( (3, 2) \)[/tex]
- [tex]\( (4, 2) \)[/tex]
4. Evaluate the objective function [tex]\( C = 2x + y \)[/tex] at these points:
- At [tex]\( (3, 2) \)[/tex]:
[tex]\[
C = 2(3) + 2 = 6 + 2 = 8
\][/tex]
- At [tex]\( (4, 2) \)[/tex]:
[tex]\[
C = 2(4) + 2 = 8 + 2 = 10
\][/tex]
5. Determine the minimum value:
Comparing the values of [tex]\( C \)[/tex] at the feasible points, we find:
[tex]\[
\min \{8, 10\} = 8
\][/tex]
Thus, the minimum value of [tex]\( C \)[/tex] is attained at the point [tex]\( (3, 2) \)[/tex], and the minimum value is:
[tex]\[
C = 8
\][/tex]
So, the optimal solution is:
[tex]\[
(x, y) = (3, 2) \quad \text{and} \quad C = 8
\][/tex]
