To solve this problem, we need to analyze the system of linear inequalities given and determine the minimum value of $z = x - 5y$. The inequalities are:

1. $4x - 3y - 3 \geq 0$
2. $x - 2y - 2 \leq 0$
3. $2x + 6y - 9 \leq 0$

First, let's rewrite the inequalities:

1. $4x - 3y \geq 3$
2. $x - 2y \leq 2$
3. $2x + 6y \leq 9$

Step-by-Step Solution:

1. Graph the inequalities to find the feasible region:

   • For $4x - 3y = 3$, rewrite as $y = \frac{4}{3}x - 1$
   • For $x - 2y = 2$, rewrite as $y = \frac{1}{2}x - 1$
   • For $2x + 6y = 9$, rewrite as $y = -\frac{1}{3}x + \frac{3}{2}$

2. Find the intersection points of the boundary lines to determine the vertices of the feasible region.

   • Intersection of $4x - 3y = 3$ and $x - 2y = 2$: 4x - 3y = 3 \\ x - 2y = 2 \end{cases}$$ Multiply the second equation by 4: $$\begin{cases} 4x - 3y = 3 \\ 4x - 8y = 8 \end{cases}$$ Subtract: $$5y = -5 \implies y = -1$$ Substitute $$y = -1$$ into $$x - 2y = 2$$: $$x - 2(-1) = 2 \implies x + 2 = 2 \implies x = 0$$ So, intersection at $$(0, -1)$$.
   • Intersection of $x - 2y = 2$ and $2x + 6y = 9$: x - 2y = 2 \\ 2x + 6y = 9 \end{cases}$$ Multiply the first equation by 2: $$\begin{cases} 2x - 4y = 4 \\ 2x + 6y = 9 \end{cases}$$ Subtract: $$10y = 5 \implies y = \frac{1}{2}$$ Substitute $$y = \frac{1}{2}$$ into $$x - 2y = 2$$: $$x - 2\left(\frac{1}{2}\right) = 2 \implies x - 1 = 2 \implies x = 3$$ So, intersection at $$(3, 0.5)$$.
   • Intersection of $4x - 3y = 3$ and $2x + 6y = 9$: 4x - 3y = 3 \\ 2x + 6y = 9 \end{cases}$$ Multiply the first equation by 2: $$\begin{cases} 8x - 6y = 6 \\ 2x + 6y = 9 \end{cases}$$ Add: $$10x = 15 \implies x = 1.5$$ Substitute $$x = 1.5$$ into $$2x + 6y = 9$$: $$2(1.5) + 6y = 9 \implies 3 + 6y = 9 \implies 6y = 6 \implies y = 1$$ So, intersection at $$(1.5, 1)$$.

3. Evaluate $z = x - 5y$ at these vertices:

   • At $(0, -1)$: $z = 0 - 5(-1) = 5$
   • At $(3, 0.5)$: $z = 3 - 5(0.5) = 3 - 2.5 = 0.5$
   • At $(1.5, 1)$: $z = 1.5 - 5(1) = 1.5 - 5 = -3.5$

The minimum value of $z = x - 5y$ is $-3.5$.

Would you like more details or have any questions?

Here are 5 questions you might want to ask next:

1. How do you graph inequalities on a coordinate plane?
2. What are the steps to find the vertices of the feasible region?
3. Can you explain how to solve a system of linear equations using substitution?
4. How do you determine the minimum or maximum value of a function subject to constraints?
5. What are other methods to solve linear programming problems?

Tip: Always check the feasibility of each intersection point within the context of all given constraints to ensure the solution is valid.