To determine which system of inequalities [tex]\((3, -7)\)[/tex] is a solution for, we will evaluate each system step-by-step.
### System A:
1. [tex]\(x + y < -4\)[/tex]
- Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex]
- [tex]\(3 + (-7) = -4\)[/tex]
- Check: [tex]\(-4 < -4\)[/tex] (False)
2. [tex]\(3x + 2y < -5\)[/tex]
- Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex]
- [tex]\(3(3) + 2(-7) = 9 - 14 = -5\)[/tex]
- Check: [tex]\(-5 < -5\)[/tex] (False)
Both inequalities must be true for the system to be true. Since the first condition is false, [tex]\( (3, -7) \)[/tex] does not satisfy system A.
### System B:
1. [tex]\(x + y \leq -4\)[/tex]
- Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex]
- [tex]\(3 + (-7) = -4\)[/tex]
- Check: [tex]\(-4 \leq -4\)[/tex] (True)
2. [tex]\(3x + 2y < -5\)[/tex]
- Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex]
- [tex]\(3(3) + 2(-7) = 9 - 14 = -5\)[/tex]
- Check: [tex]\(-5 < -5\)[/tex] (False)
Both inequalities must be true for the system to be true. Since the second condition is false, [tex]\( (3, -7) \)[/tex] does not satisfy system B.
### System C:
1. [tex]\(x + y < -4\)[/tex]
- Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex]
- [tex]\(3 + (-7) = -4\)[/tex]
- Check: [tex]\(-4 < -4\)[/tex] (False)
2. [tex]\(3x + 2y \leq -5\)[/tex]
- Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex]
- [tex]\(3(3) + 2(-7) = 9 - 14 = -5\)[/tex]
- Check: [tex]\(-5 \leq -5\)[/tex] (True)
Both inequalities must be true for the system to be true. Since the first condition is false, [tex]\( (3, -7) \)[/tex] does not satisfy system C.
### System D:
1. [tex]\(x + y \leq -4\)[/tex]
- Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex]
- [tex]\(3 + (-7) = -4\)[/tex]
- Check: [tex]\(-4 \leq -4\)[/tex] (True)
2. [tex]\(3x + 2y \leq -5\)[/tex]
- Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex]
- [tex]\(3(3) + 2(-7) = 9 - 14 = -5\)[/tex]
- Check: [tex]\(-5 \leq -5\)[/tex] (True)
Both inequalities are true. Therefore, [tex]\((3, -7)\)[/tex] satisfies system D.
### Conclusion:
The point [tex]\((3, -7)\)[/tex] is a solution to system D.
So, the correct answer is:
D.
[tex]\[
\begin{array}{l}
x+y \leq-4 \\
3 x+2 y \leq-5
\end{array}
\][/tex]
