Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set.

To solve the system of inequalities graphically, we can approach it in the following way:

1. Graph each inequality.

   • For \( y > -3x - 3 \):

     • First graph the line \( y = -3x - 3 \). This line has a slope of -3 and a y-intercept of -3. Because it is a 'greater than' inequality, we will use a dashed line (indicating that points on the line are not included in the solution) and shade the area above the line.

   • For \( y > x - 7 \):

     • Next, graph the line \( y = x - 7 \). This line has a slope of 1 and a y-intercept of -7. Similar to the first line, we will use a dashed line here as well and shade the area above this line.

2. Determine the overlapping region:

   • The solution to the system of inequalities will be the region that is shaded for both inequalities. This is the area where both conditions are satisfied: above the line \( y = -3x - 3 \) and above the line \( y = x - 7 \).

3. Finding a point within the shaded region:

   • We can easily find a point in the overlapping shaded region. For example, let’s take \( (0, 0) \):
     • For \( y > -3x - 3 \): \[ 0 > -3(0) - 3 \implies 0 > -3 \quad \text{(True)} \]
     • For \( y > x - 7 \): \[ 0 > 0 - 7 \implies 0 > -7 \quad \text{(True)} \]
     • The point \( (0, 0) \) satisfies both inequalities, so it is within the solution set.

Conclusion:

The shaded area where both inequalities are true represents the solution set. A valid point in the solution set is \( \boxed{(0, 0)} \).

Let’s carefully go through the solution step by step again.

Step 1: Graph the inequalities

1. For \( y > -3x - 3 \):

   • The line \( y = -3x - 3 \) has a y-intercept at \(-3\) and a slope of \(-3\).
   • To graph this line:
     • When \( x = 0 \), \( y = -3 \): the point is \( (0, -3) \).
     • When \( x = 1 \), \( y = -6 \): the point is \( (1, -6) \).
   • Draw a dashed line through these points, indicating that points on the line are not included in the solution.
   • Shade the area above this line (where \( y \) is greater).

2. For \( y > x - 7 \):

   • The line \( y = x - 7 \) has a y-intercept at \(-7\) and a slope of \(1\).
   • To graph this line:
     • When \( x = 0 \), \( y = -7 \): the point is \( (0, -7) \).
     • When \( x = 7 \), \( y = 0 \): the point is \( (7, 0) \).
   • Draw a dashed line through these points and shade the area above this line.

Step 2: Identify the solution region

• The solution set is the region where both shaded areas overlap. We are looking for the area that is above both lines.

Step 3: Find a valid point in the solution region

To find a point in the shaded area where both inequalities hold true, we can consider some test points.

Choosing a test point

For a quick check, we can try \( (8, 2) \):

• Check \( y > -3x - 3 \): \[ 2 > -3(8) - 3 \implies 2 > -24 - 3 \implies 2 > -27 \quad \text{(True)} \]

• Check \( y > x - 7 \): \[ 2 > 8 - 7 \implies 2 > 1 \quad \text{(True)} \]

Both inequalities are satisfied, so \( (8, 2) \) is a valid point in the solution set.

Final Conclusion

The correct coordinates of a point in the solution set is \( \boxed{(8, 2)} \).

If you have graphing capabilities or software, it will help visualize and confirm this solution.