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Sagot :
To determine for which system of inequalities the point [tex]\((3, -7)\)[/tex] is a solution, we will check each system individually.
First, let's assign the values of the point to the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = 3 \][/tex]
[tex]\[ y = -7 \][/tex]
Now let's examine each system:
System A:
[tex]\[ \begin{array}{l} x + y < -4 \\ 3x + 2y < -5 \end{array} \][/tex]
Substituting [tex]\( x = 3 \)[/tex] and [tex]\( y = -7 \)[/tex] into the inequalities:
[tex]\[ 3 + (-7) < -4 \][/tex]
[tex]\[ -4 < -4 \quad \text{(False)} \][/tex]
For the second inequality:
[tex]\[ 3(3) + 2(-7) < -5 \][/tex]
[tex]\[ 9 - 14 < -5 \][/tex]
[tex]\[ -5 < -5 \quad \text{(False)} \][/tex]
Since the first inequality is false, [tex]\((3, -7)\)[/tex] is not a solution to System A.
System B:
[tex]\[ \begin{array}{l} x + y \leq -4 \\ 3x + 2y < -5 \end{array} \][/tex]
Substituting [tex]\( x = 3 \)[/tex] and [tex]\( y = -7 \)[/tex] into the inequalities:
[tex]\[ 3 + (-7) \leq -4 \][/tex]
[tex]\[ -4 \leq -4 \quad \text{(True)} \][/tex]
For the second inequality:
[tex]\[ 3(3) + 2(-7) < -5 \][/tex]
[tex]\[ 9 - 14 < -5 \][/tex]
[tex]\[ -5 < -5 \quad \text{(False)} \][/tex]
Since the second inequality is false, [tex]\((3, -7)\)[/tex] is not a solution to System B.
System C:
[tex]\[ \begin{array}{l} x + y < -4 \\ 3x + 2y \leq -5 \end{array} \][/tex]
Substituting [tex]\( x = 3 \)[/tex] and [tex]\( y = -7 \)[/tex] into the inequalities:
[tex]\[ 3 + (-7) < -4 \][/tex]
[tex]\[ -4 < -4 \quad \text{(False)} \][/tex]
For the second inequality:
[tex]\[ 3(3) + 2(-7) \leq -5 \][/tex]
[tex]\[ 9 - 14 \leq -5 \][/tex]
[tex]\[ -5 \leq -5 \quad \text{(True)} \][/tex]
Since the first inequality is false, [tex]\((3, -7)\)[/tex] is not a solution to System C.
System D:
[tex]\[ \begin{array}{l} x + y \leq -4 \\ 3x + 2y \leq -5 \end{array} \][/tex]
Substituting [tex]\( x = 3 \)[/tex] and [tex]\( y = -7 \)[/tex] into the inequalities:
[tex]\[ 3 + (-7) \leq -4 \][/tex]
[tex]\[ -4 \leq -4 \quad \text{(True)} \][/tex]
For the second inequality:
[tex]\[ 3(3) + 2(-7) \leq -5 \][/tex]
[tex]\[ 9 - 14 \leq -5 \][/tex]
[tex]\[ -5 \leq -5 \quad \text{(True)} \][/tex]
Since both inequalities are true, [tex]\((3, -7)\)[/tex] is a solution to System D.
Hence, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
First, let's assign the values of the point to the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = 3 \][/tex]
[tex]\[ y = -7 \][/tex]
Now let's examine each system:
System A:
[tex]\[ \begin{array}{l} x + y < -4 \\ 3x + 2y < -5 \end{array} \][/tex]
Substituting [tex]\( x = 3 \)[/tex] and [tex]\( y = -7 \)[/tex] into the inequalities:
[tex]\[ 3 + (-7) < -4 \][/tex]
[tex]\[ -4 < -4 \quad \text{(False)} \][/tex]
For the second inequality:
[tex]\[ 3(3) + 2(-7) < -5 \][/tex]
[tex]\[ 9 - 14 < -5 \][/tex]
[tex]\[ -5 < -5 \quad \text{(False)} \][/tex]
Since the first inequality is false, [tex]\((3, -7)\)[/tex] is not a solution to System A.
System B:
[tex]\[ \begin{array}{l} x + y \leq -4 \\ 3x + 2y < -5 \end{array} \][/tex]
Substituting [tex]\( x = 3 \)[/tex] and [tex]\( y = -7 \)[/tex] into the inequalities:
[tex]\[ 3 + (-7) \leq -4 \][/tex]
[tex]\[ -4 \leq -4 \quad \text{(True)} \][/tex]
For the second inequality:
[tex]\[ 3(3) + 2(-7) < -5 \][/tex]
[tex]\[ 9 - 14 < -5 \][/tex]
[tex]\[ -5 < -5 \quad \text{(False)} \][/tex]
Since the second inequality is false, [tex]\((3, -7)\)[/tex] is not a solution to System B.
System C:
[tex]\[ \begin{array}{l} x + y < -4 \\ 3x + 2y \leq -5 \end{array} \][/tex]
Substituting [tex]\( x = 3 \)[/tex] and [tex]\( y = -7 \)[/tex] into the inequalities:
[tex]\[ 3 + (-7) < -4 \][/tex]
[tex]\[ -4 < -4 \quad \text{(False)} \][/tex]
For the second inequality:
[tex]\[ 3(3) + 2(-7) \leq -5 \][/tex]
[tex]\[ 9 - 14 \leq -5 \][/tex]
[tex]\[ -5 \leq -5 \quad \text{(True)} \][/tex]
Since the first inequality is false, [tex]\((3, -7)\)[/tex] is not a solution to System C.
System D:
[tex]\[ \begin{array}{l} x + y \leq -4 \\ 3x + 2y \leq -5 \end{array} \][/tex]
Substituting [tex]\( x = 3 \)[/tex] and [tex]\( y = -7 \)[/tex] into the inequalities:
[tex]\[ 3 + (-7) \leq -4 \][/tex]
[tex]\[ -4 \leq -4 \quad \text{(True)} \][/tex]
For the second inequality:
[tex]\[ 3(3) + 2(-7) \leq -5 \][/tex]
[tex]\[ 9 - 14 \leq -5 \][/tex]
[tex]\[ -5 \leq -5 \quad \text{(True)} \][/tex]
Since both inequalities are true, [tex]\((3, -7)\)[/tex] is a solution to System D.
Hence, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
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