To determine the solution set graphically for the system of inequalities, we need to:

1. Plot each inequality line and shade the appropriate region.
2. Identify the region where all inequalities overlap, as this represents the solution set.

System of Inequalities

1. $x + y \leq 5$
2. $2x + y \leq 7$
3. $2x - y \geq -1$
4. $x \geq 0$
5. $y \geq 0$

Steps to Graph Each Inequality

1. Graph $x + y = 5$:

   • This is a line with a slope of -1 and y-intercept at $(0, 5)$.
   • Another point on the line is $(5, 0)$.
   • Shade the region below this line for $x + y \leq 5$.

2. Graph $2x + y = 7$:

   • This is a line with a slope of -2 and y-intercept at $(0, 7)$.
   • Another point on the line is $(3.5, 0)$.
   • Shade the region below this line for $2x + y \leq 7$.

3. Graph $2x - y = -1$:

   • Rearrange this to $y = 2x + 1$.
   • This line has a slope of 2 and y-intercept at $(0, 1)$.
   • Shade the region above this line for $2x - y \geq -1$.

4. Graph $x \geq 0$ and $y \geq 0$:

   • These inequalities restrict the solution to the first quadrant.

Finding the Feasible Region

The solution set is where all shaded regions overlap, and it lies within the first quadrant due to $x \geq 0$ and $y \geq 0$. The boundary points of this region can be found by determining intersections of the lines.

Would you like further assistance with finding these intersection points or visualizing the graph?

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Related Questions:

1. What are the boundary points of the feasible region?
2. How can we solve the inequalities algebraically to verify the graphical solution?
3. How do we determine the vertices of the solution region?
4. What are real-world applications of solving systems of inequalities graphically?
5. How does changing one inequality affect the solution region?

Tip: When plotting inequalities, always start by graphing each line first, then shade regions and check for overlaps.