To determine how many solutions the system of equations has, let's analyze the two equations:

1. $y = -x - 2$
2. $y = x + 7$

These are linear equations, and the solution to the system of equations can be found by identifying if and where the lines intersect.

Steps:

1. Graphing the Equations:

   • The first equation, $y = -x - 2$, has a slope of $-1$ and a y-intercept of $-2$.
   • The second equation, $y = x + 7$, has a slope of $1$ and a y-intercept of $7$.

2. Analyzing Slope and Intersection:

   • Since the slopes are different ($-1$ and $1$), the lines will intersect at exactly one point, because lines with different slopes are not parallel and intersect at a unique point in a two-dimensional plane.

3. Conclusion:

   • Since the lines intersect at only one point, the system of equations has one solution.

Answer

The correct answer is:

• One solution

Would you like further details on any part of this process, or have any questions?

Here are some related questions for further exploration:

1. What would happen if the lines had the same slope but different y-intercepts?
2. How can we find the exact point of intersection algebraically?
3. What does it mean for a system of equations to have no solutions?
4. How does changing the slope affect the number of solutions in a system of linear equations?
5. What is the geometric interpretation of systems with infinitely many solutions?

Tip: When two lines have different slopes, they will always intersect at one unique point in a 2D coordinate plane, indicating a single solution for the system.