To find the minimum value of the expression [tex]\( C = -2x + y \)[/tex] given the constraints:
[tex]\[
\begin{aligned}
& -5 \leq x \leq 4 \\
& -1 \leq y \leq 3
\end{aligned}
\][/tex]
Let's explore the function [tex]\( C \)[/tex] over the feasible region defined by these constraints. This involves checking the values of [tex]\( C \)[/tex] at each possible combination of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in their respective ranges.
1. List the possible values for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
- [tex]\( x \)[/tex] can take any integer value from [tex]\(-5\)[/tex] to [tex]\(4\)[/tex].
- [tex]\( y \)[/tex] can take any integer value from [tex]\(-1\)[/tex] to [tex]\(3\)[/tex].
2. For each combination of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] within these ranges, compute [tex]\( C \)[/tex]:
We'll evaluate [tex]\( C \)[/tex] for all pairs [tex]\((x, y)\)[/tex]:
- When [tex]\( x = -5 \)[/tex]:
- For [tex]\( y = -1 \)[/tex], [tex]\( C = -2(-5) + (-1) = 10 - 1 = 9 \)[/tex]
- For [tex]\( y = 0 \)[/tex], [tex]\( C = -2(-5) + 0 = 10 \)[/tex]
- For [tex]\( y = 1 \)[/tex], [tex]\( C = -2(-5) + 1 = 11 \)[/tex]
- For [tex]\( y = 2 \)[/tex], [tex]\( C = -2(-5) + 2 = 12 \)[/tex]
- For [tex]\( y = 3 \)[/tex], [tex]\( C = -2(-5) + 3 = 13 \)[/tex]
- When [tex]\( x = -4 \)[/tex]:
- For [tex]\( y = -1 \)[/tex], [tex]\( C = -2(-4) + (-1) = 8 - 1 = 7 \)[/tex]
- For [tex]\( y = 0 \)[/tex], [tex]\( C = -2(-4) + 0 = 8 \)[/tex]
- For [tex]\( y = 1 \)[/tex], [tex]\( C = -2(-4) + 1 = 9 \)[/tex]
- For [tex]\( y = 2 \)[/tex], [tex]\( C = -2(-4) + 2 = 10 \)[/tex]
- For [tex]\( y = 3 \)[/tex], [tex]\( C = -2(-4) + 3 = 11 \)[/tex]
- Continue this process for each value of [tex]\( x \)[/tex]:
- ...
- When [tex]\( x = 4 \)[/tex]:
- For [tex]\( y = -1 \)[/tex], [tex]\( C = -2(4) + (-1) = -8 - 1 = -9 \)[/tex]
- For [tex]\( y = 0 \)[/tex], [tex]\( C = -2(4) + 0 = -8 \)[/tex]
- For [tex]\( y = 1 \)[/tex], [tex]\( C = -2(4) + 1 = -7 \)[/tex]
- For [tex]\( y = 2 \)[/tex], [tex]\( C = -2(4) + 2 = -6 \)[/tex]
- For [tex]\( y = 3 \)[/tex], [tex]\( C = -2(4) + 3 = -5 \)[/tex]
3. Identify the minimum value of [tex]\( C \)[/tex] among all the computed values:
From the evaluation, the smallest value of [tex]\( C \)[/tex] is [tex]\(-9\)[/tex].
4. Determine the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that yield this minimum value:
The minimum value [tex]\( C = -9 \)[/tex] is achieved at [tex]\( x = 4 \)[/tex] and [tex]\( y = -1 \)[/tex].
Thus, the minimum value of [tex]\( C \)[/tex] given the constraints is:
[tex]\[
C = -9 \quad \text{when} \quad (x, y) = (4, -1)
\][/tex]
