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

1. Graph the Constraints: Plot the three inequalities to determine the feasible region.
2. Find the Corner Points: Identify the coordinates of the corner points of the feasible region.
3. Evaluate the Objective Function at Each Corner: Calculate $C = -10x + 10y$ at each of the corner points to find the minimum value.

Step 1: Graph the Constraints

The constraints are:

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

These describe half-planes, and the feasible region is the intersection of all these half-planes. We will first express the equations of the lines:

1. $y = x + 2$
2. $y = -3x + 10$
3. $y = -x + 4$

These are straight lines, and we need to determine which side of each line satisfies the inequality.

Step 2: Find the Intersection Points (Corner Points)

We can find the intersections by solving pairs of equations:

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

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

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

Step 3: Evaluate the Objective Function at Each Corner

Now we evaluate the objective function $C = -10x + 10y$ at each of the corner points $(2, 4)$, $(1, 3)$, and $(3, 1)$:

1. At $(2, 4)$: $C = -10(2) + 10(4) = -20 + 40 = 20$
2. At $(1, 3)$: $C = -10(1) + 10(3) = -10 + 30 = 20$
3. At $(3, 1)$: $C = -10(3) + 10(1) = -30 + 10 = -20$

Step 4: Conclusion

The objective function $C = -10x + 10y$ achieves its minimum value at the point $(3, 1)$, where $C = -20$.

Final Answer:

The values of $x$ and $y$ that minimize the objective function are $\boxed{x = 3}$ and $\boxed{y = 1}$.

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Would you like further details on any part of the process?

Here are some related questions to explore:

1. How do you graph a system of inequalities to find the feasible region?
2. What are corner points in a linear programming problem and why are they important?
3. How does the simplex method differ from graphical methods in linear programming?
4. Can you solve a linear programming problem without graphing?
5. What happens if the objective function is maximized instead of minimized?

Tip: Always ensure that the intersection points are within the feasible region defined by the inequalities before evaluating the objective function.