To solve the system of equations given in the image:

Equations:

1. $y = -(x - 4)^2 + 9$ (a downward-opening parabola with vertex at $(4, 9)$)
2. $y = 2x + 2$ (a linear equation with slope 2 and y-intercept at $(0, 2)$).

Step 1: Analyze and Solve Graphically

By graphing both equations, we identify the points of intersection. These represent the solutions to the system of equations. From the graph in the image, the solutions are:

• $(1, 4)$
• $(6, 5)$
• $(3, 8)$

Step 2: Verify Each Solution

To confirm, substitute these points into both equations:

1. For $(1, 4)$:

   • $y = -(1 - 4)^2 + 9 = -(-3)^2 + 9 = -9 + 9 = 4$ (True)
   • $y = 2(1) + 2 = 2 + 2 = 4$ (True)

2. For $(6, 5)$:

   • $y = -(6 - 4)^2 + 9 = -(2)^2 + 9 = -4 + 9 = 5$ (True)
   • $y = 2(6) + 2 = 12 + 2 = 5$ (True)

3. For $(3, 8)$:

   • $y = -(3 - 4)^2 + 9 = -(1)^2 + 9 = -1 + 9 = 8$ (True)
   • $y = 2(3) + 2 = 6 + 2 = 8$ (True)

Final Answer:

The solutions are $(1, 4)$, $(6, 5)$, and $(3, 8)$.

Would you like further clarification or details? Here are five related questions for further exploration:

1. How can you find the vertex of a parabola from its equation?
2. What is the algebraic method to solve this system of equations instead of graphing?
3. How do you determine whether a system of equations has one, two, or no solutions?
4. Can the slope of the linear equation affect the number of solutions with a quadratic equation?
5. What are the key differences between graphing and substitution methods for solving systems?

Tip: Always verify solutions by substituting them back into both equations to avoid errors.