Let's work through the problem step by step. You’ve provided a linear programming problem with these constraints and objective function:

Constraints:

1. $y \leq -\frac{1}{2}x + 6$
2. $y \leq \frac{1}{2}x$
3. $y \geq \frac{1}{4}x$

Objective function:

$f(x, y) = 7x - 5y$

Steps to graph the system of constraints:

1. Plot each of the lines corresponding to the constraints.
   These lines will be in the form of linear equations, and we need to graph them on the coordinate plane:

   • $y = -\frac{1}{2}x + 6$
   • $y = \frac{1}{2}x$
   • $y = \frac{1}{4}x$

2. Shade the appropriate regions based on the inequalities:

   • For $y \leq -\frac{1}{2}x + 6$, shade the region below the line.
   • For $y \leq \frac{1}{2}x$, shade the region below the line.
   • For $y \geq \frac{1}{4}x$, shade the region above the line.

The feasible region will be where all the shaded regions overlap.

Would you like me to graph this for you? I can also help identify which line to change or analyze based on your specific needs.