To determine the number of solutions to the given system of equations, we need to find the points of intersection between the two equations:
[tex]\[
\left\{
\begin{array}{l}
y = -(x - 3)^2 + 3 \\
y = x - 6
\end{array}
\right.
\][/tex]
By setting the two equations equal to each other, we can solve for [tex]\( x \)[/tex]:
[tex]\[
-(x - 3)^2 + 3 = x - 6
\][/tex]
Let's solve this equation step by step.
1. Expand [tex]\( -(x - 3)^2 \)[/tex]:
[tex]\[
-(x - 3)^2 = -((x - 3)(x - 3)) = -(x^2 - 6x + 9) = -x^2 + 6x - 9
\][/tex]
2. Substitute the expansion into the equation:
[tex]\[
-x^2 + 6x - 9 + 3 = x - 6
\][/tex]
3. Simplify the equation:
[tex]\[
-x^2 + 6x - 6 = x - 6
\][/tex]
4. Move all terms to one side to set the equation to zero:
[tex]\[
-x^2 + 6x - 6 - x + 6 = 0
\][/tex]
[tex]\[
-x^2 + 5x = 0
\][/tex]
5. Factor the equation:
[tex]\[
x(-x + 5) = 0
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 0 \quad \text{or} \quad x = 5
\][/tex]
Now we have two values for [tex]\( x \)[/tex]: [tex]\( x = 0 \)[/tex] and [tex]\( x = 5 \)[/tex]. We need to find the corresponding [tex]\( y \)[/tex] values for each [tex]\( x \)[/tex] to get the points of intersection.
For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 0 - 6 = -6
\][/tex]
So the point is [tex]\((0, -6)\)[/tex].
For [tex]\( x = 5 \)[/tex]:
[tex]\[
y = 5 - 6 = -1
\][/tex]
So the point is [tex]\((5, -1)\)[/tex].
Therefore, the solutions to the system of equations are:
[tex]\[
(0,-6) \quad \text{and} \quad (5,-1)
\][/tex]
Hence, the number of solutions to the given system is two, and they are:
[tex]\[
\boxed{(0,-6)} \quad \text{and} \quad \boxed{(5,-1)}
\][/tex]
