To determine which ordered pair is a solution to the system of inequalities:
[tex]\[
\begin{cases}
y \leq -x^2 + 8x \\
y > x^2 - 3
\end{cases}
\][/tex]
We must check each pair to see if it satisfies both inequalities.
### Pair 1: [tex]\((-1, 4)\)[/tex]
1. Check [tex]\( y \leq -x^2 + 8x \)[/tex]:
[tex]\( y = 4 \)[/tex]
[tex]\( x = -1 \)[/tex]
[tex]\[
-(-1)^2 + 8(-1) = -1 - 8 = -9
\][/tex]
[tex]\[
4 \leq -9 \quad \text{(False)}
\][/tex]
### Pair 2: [tex]\((0, 9)\)[/tex]
1. Check [tex]\( y \leq -x^2 + 8x \)[/tex]:
[tex]\( y = 9 \)[/tex]
[tex]\( x = 0 \)[/tex]
[tex]\[
-0^2 + 8(0) = 0
\][/tex]
[tex]\[
9 \leq 0 \quad \text{(False)}
\][/tex]
### Pair 3: [tex]\((2, 7)\)[/tex]
1. Check [tex]\( y \leq -x^2 + 8x \)[/tex]:
[tex]\( y = 7 \)[/tex]
[tex]\( x = 2 \)[/tex]
[tex]\[
-(2)^2 + 8(2) = -4 + 16 = 12
\][/tex]
[tex]\[
7 \leq 12 \quad \text{(True)}
\][/tex]
2. Check [tex]\( y > x^2 - 3 \)[/tex]:
[tex]\( y = 7 \)[/tex]
[tex]\[
(2)^2 - 3 = 4 - 3 = 1
\][/tex]
[tex]\[
7 > 1 \quad \text{(True)}
\][/tex]
Since both conditions are satisfied, [tex]\((2, 7)\)[/tex] is a solution.
### Pair 4: [tex]\((4, 3)\)[/tex]
1. Check [tex]\( y \leq -x^2 + 8x \)[/tex]:
[tex]\( y = 3 \)[/tex]
[tex]\( x = 4 \)[/tex]
[tex]\[
-(4)^2 + 8(4) = -16 + 32 = 16
\][/tex]
[tex]\[
3 \leq 16 \quad \text{(True)}
\][/tex]
2. Check [tex]\( y > x^2 - 3 \)[/tex]:
[tex]\( y = 3 \)[/tex]
[tex]\[
(4)^2 - 3 = 16 - 3 = 13
\][/tex]
[tex]\[
3 > 13 \quad \text{(False)}
\][/tex]
Since [tex]\((4, 3)\)[/tex] does not satisfy the second inequality, it is not a solution.
Thus, the ordered pair that satisfies both inequalities is [tex]\((2, 7)\)[/tex].
The correct answer is:
[tex]\[
(2, 7)
\][/tex]
