To minimize the objective function $C = -10x + 10y$ subject to the constraints:

1. $y \leq x + 2$
2. $y \leq -3x + 10$
3. $y \geq -x + 4$

We will follow these steps:

Step 1: Graph the constraints to find the feasible region.

We first graph the lines of each inequality to visualize the feasible region, as the solution will lie at a vertex of this region.

Graph the first constraint $y \leq x + 2$:

This represents a line $y = x + 2$, which has slope 1 and y-intercept 2. The region is below this line.

Graph the second constraint $y \leq -3x + 10$:

This represents the line $y = -3x + 10$, with slope -3 and y-intercept 10. The region is below this line.

Graph the third constraint $y \geq -x + 4$:

This represents the line $y = -x + 4$, with slope -1 and y-intercept 4. The region is above this line.

Step 2: Find the intersection points of the lines.

These points are where the constraints intersect, and they will form the vertices of the feasible region.

• Intersection of $y = x + 2$ and $y = -3x + 10$: $x + 2 = -3x + 10$ $4x = 8 \quad \Rightarrow \quad x = 2$ Substituting $x = 2$ into $y = x + 2$: $y = 2 + 2 = 4$ So, the intersection point is $(2, 4)$.

• Intersection of $y = x + 2$ and $y = -x + 4$: $x + 2 = -x + 4$ $2x = 2 \quad \Rightarrow \quad x = 1$ Substituting $x = 1$ into $y = x + 2$: $y = 1 + 2 = 3$ So, the intersection point is $(1, 3)$.

• Intersection of $y = -3x + 10$ and $y = -x + 4$: $-3x + 10 = -x + 4$ $-2x = -6 \quad \Rightarrow \quad x = 3$ Substituting $x = 3$ into $y = -3x + 10$: $y = -3(3) + 10 = -9 + 10 = 1$ So, the intersection point is $(3, 1)$.

Step 3: Evaluate the objective function at each vertex.

Now, evaluate $C = -10x + 10y$ at each of the intersection points:

• At $(2, 4)$: $C = -10(2) + 10(4) = -20 + 40 = 20$

• At $(1, 3)$: $C = -10(1) + 10(3) = -10 + 30 = 20$

• At $(3, 1)$: $C = -10(3) + 10(1) = -30 + 10 = -20$

Step 4: Identify the minimum value.

The minimum value of $C$ is $-20$, which occurs at the point $(3, 1)$.

Final Answer:

The values of $x$ and $y$ that minimize the objective function are: $x = 3, \quad y = 1$

Would you like further clarification or details on any of the steps? Here's a list of related questions that might be helpful:

1. How can we determine if a given point lies within the feasible region?
2. What would happen if the objective function were changed to a maximization problem instead of minimization?
3. How do we handle cases where constraints are non-linear in linear programming problems?
4. What would be the result if one of the constraints were to be modified?
5. How can we verify the feasibility of a solution using the Simplex method?

Tip: Always ensure that all the constraints are satisfied at the point you are checking for the objective function to ensure that it's part of the feasible region.