To determine which ordered pair is a solution to the system of inequalities:
[tex]\[
\begin{array}{l}
y \leq -x^2 + 8x \\
y > x^2 - 3
\end{array}
\][/tex]
we need to check each given ordered pair against both inequalities.
Step-by-Step Checking:
1. Ordered Pair [tex]\((-1, 4)\)[/tex]:
- Check [tex]\( y \leq -x^2 + 8x \)[/tex]:
[tex]\[
4 \leq -(-1)^2 + 8(-1) \implies 4 \leq -1 - 8 \implies 4 \leq -9
\][/tex]
This is false.
- Since it does not satisfy the first inequality, we don't need to check the second inequality for this pair.
2. Ordered Pair [tex]\((0, 9)\)[/tex]:
- Check [tex]\( y \leq -x^2 + 8x \)[/tex]:
[tex]\[
9 \leq -(0)^2 + 8(0) \implies 9 \leq 0
\][/tex]
This is false.
- Since it does not satisfy the first inequality, we don't need to check the second inequality for this pair.
3. Ordered Pair [tex]\((2, 7)\)[/tex]:
- Check [tex]\( y \leq -x^2 + 8x \)[/tex]:
[tex]\[
7 \leq -(2)^2 + 8(2) \implies 7 \leq -4 + 16 \implies 7 \leq 12
\][/tex]
This is true.
- Check [tex]\( y > x^2 - 3 \)[/tex]:
[tex]\[
7 > (2)^2 - 3 \implies 7 > 4 - 3 \implies 7 > 1
\][/tex]
This is also true.
Since both inequalities are satisfied, [tex]\((2, 7)\)[/tex] is a solution to the system.
4. Ordered Pair [tex]\((4, 3)\)[/tex]:
- Check [tex]\( y \leq -x^2 + 8x \)[/tex]:
[tex]\[
3 \leq -(4)^2 + 8(4) \implies 3 \leq -16 + 32 \implies 3 \leq 16
\][/tex]
This is true.
- Check [tex]\( y > x^2 - 3 \)[/tex]:
[tex]\[
3 > (4)^2 - 3 \implies 3 > 16 - 3 \implies 3 > 13
\][/tex]
This is false.
- Since it does not satisfy the second inequality, [tex]\((4, 3)\)[/tex] is not a solution.
Based on the evaluations, the ordered pair [tex]\((2, 7)\)[/tex] satisfies both inequalities. Thus, the ordered pair that is a solution to the given system of inequalities is:
[tex]\(\boxed{(2, 7)}\)[/tex]
