The given linear programming problem is:

Maximize: $z = 21x + 11y$

Subject to:

1. $3x + 5y \leq 15$
2. $7x + y \leq 15$
3. $x \geq 0, \, y \geq 0$

Step 1: Graph the constraints

We will first convert the inequalities to equalities and find the boundary points.

1. For $3x + 5y = 15$:

   • When $x = 0$, $y = 3$.
   • When $y = 0$, $x = 5$. Thus, this line passes through $(5, 0)$ and $(0, 3)$.

2. For $7x + y = 15$:

   • When $x = 0$, $y = 15$.
   • When $y = 0$, $x = \frac{15}{7} \approx 2.14$. Thus, this line passes through $(2.14, 0)$ and $(0, 15)$.

Step 2: Find the feasible region

Plotting the two lines on a graph, along with the conditions $x \geq 0$ and $y \geq 0$, the feasible region is the area where all constraints are satisfied. The vertices of this region are the points where the boundary lines intersect or hit the axes.

Step 3: Identify the corner points

The corner points of the feasible region are:

1. $(0, 0)$
2. $(5, 0)$
3. $(0, 3)$
4. The intersection of the lines $3x + 5y = 15$ and $7x + y = 15$.

Finding the intersection of the lines:

Solve the system of equations:

1. $3x + 5y = 15$
2. $7x + y = 15$

Multiply the second equation by 5 to eliminate $y$:

$7x + y = 15 \quad \Rightarrow \quad 35x + 5y = 75$

Subtract the first equation from this:

$(35x + 5y) - (3x + 5y) = 75 - 15$ $32x = 60 \quad \Rightarrow \quad x = \frac{60}{32} = 1.875$

Substitute $x = 1.875$ into $7x + y = 15$:

$7(1.875) + y = 15 \quad \Rightarrow \quad 13.125 + y = 15 \quad \Rightarrow \quad y = 1.875$

Thus, the intersection point is $(1.875, 1.875)$.

Step 4: Evaluate the objective function

Evaluate $z = 21x + 11y$ at each corner point:

1. At $(0, 0)$: $z = 21(0) + 11(0) = 0$
2. At $(5, 0)$: $z = 21(5) + 11(0) = 105$
3. At $(0, 3)$: $z = 21(0) + 11(3) = 33$
4. At $(1.875, 1.875)$: $z = 21(1.875) + 11(1.875) = 39.375 + 20.625 = 60$

Step 5: Conclusion

The maximum value of $z = 105$ occurs at the point $(5, 0)$.

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Would you like more details on any of the steps?

Here are 5 follow-up questions that might be of interest:

1. How do we interpret the feasible region in linear programming?
2. Can you explain how the graphical method works for linear programming problems?
3. What happens when the maximum point lies on the intersection of two boundary lines?
4. How can we use the simplex method to solve this problem?
5. What is the difference between linear and nonlinear programming?

Tip: In linear programming, the maximum or minimum value of the objective function always occurs at one of the vertices (corner points) of the feasible region.