To solve this problem, we need to set up a system of linear inequalities based on the constraints and then solve them graphically. Let's go through the process:

Step 1: Define the Variables

Let:

• $x$ = the number of product X produced
• $y$ = the number of product Y produced

Step 2: Define the Constraints

We are given the following information about the resources and the products:

1. Machine Time:

   • Product X requires 2 hours of machine time.
   • Product Y requires 1 hour of machine time.
   • The total machine time available is 100 hours.

   The equation for machine time is: $2x + y \leq 100$

2. Labor Hours:

   • Product X requires 1 hour of labor.
   • Product Y requires 3 hours of labor.
   • The total labor hours available is 90 hours.

   The equation for labor hours is: $x + 3y \leq 90$

3. Non-negativity: The number of products produced cannot be negative, so: $x \geq 0 \quad \text{and} \quad y \geq 0$

Step 3: Set Up the Objective Function

The objective is to maximize the profit, which is given by: $P = 30x + 20y$ This is the profit equation, but the goal is to find the feasible region by solving the system of constraints first.

Step 4: Graphing the Constraints

We need to find the point where the constraints intersect.

1. Graph the first constraint: $2x + y = 100$ Solve for $y$ in terms of $x$: $y = 100 - 2x$ This is a straight line with a y-intercept of 100 and a slope of -2.

2. Graph the second constraint: $x + 3y = 90$ Solve for $y$ in terms of $x$: $y = \frac{90 - x}{3}$ This is another straight line with a y-intercept of 30 and a slope of -$\frac{1}{3}$.

Step 5: Find the Intersection Point

To find where the two lines intersect, set the two equations for $y$ equal to each other: $100 - 2x = \frac{90 - x}{3}$

Multiply both sides by 3 to eliminate the fraction: $3(100 - 2x) = 90 - x$ $300 - 6x = 90 - x$ Now, solve for $x$: $300 - 90 = 6x - x$ $210 = 5x$ $x = 42$

Substitute $x = 42$ into either equation for $y$. Using the first equation: $y = 100 - 2(42) = 100 - 84 = 16$

Step 6: Conclusion

The two lines intersect at the point $(42, 16)$. Therefore, the optimal number of products to produce is:

• 42 units of product X
• 16 units of product Y

This is the point where the two constraints are satisfied simultaneously.

Would you like further details on how to graph this or more information about the optimization process? Here are some related questions you might find interesting:

1. How do you identify the feasible region in linear programming problems?
2. How do you calculate the maximum profit using the objective function?
3. What is the role of the corner points in graphical optimization?
4. How do you handle inequalities when solving linear programming problems graphically?
5. What would happen if the available resources (machine time or labor hours) increased?

Tip: When solving linear programming problems graphically, it’s important to plot the constraints as lines and identify the feasible region formed by the intersection of these lines. The optimal solution is usually found at one of the corner points of this region.