To determine which point is a solution to the given system of inequalities, we will evaluate each point against both inequalities:
[tex]\[
\begin{cases}
-2x + 3y \geq 1 \\
-5x + 6y \leq 1
\end{cases}
\][/tex]
Let's check each point one-by-one:
1. Point (6, 6):
For the first inequality, substitute [tex]\( x = 6 \)[/tex] and [tex]\( y = 6 \)[/tex]:
[tex]\[
-2(6) + 3(6) = -12 + 18 = 6 \quad (\text{which is } \geq 1)
\][/tex]
This point satisfies the first inequality.
For the second inequality, substitute [tex]\( x = 6 \)[/tex] and [tex]\( y = 6 \)[/tex]:
[tex]\[
-5(6) + 6(6) = -30 + 36 = 6 \quad (\text{which is not } \leq 1)
\][/tex]
This point does not satisfy the second inequality.
2. Point (7, 8):
For the first inequality, substitute [tex]\( x = 7 \)[/tex] and [tex]\( y = 8 \)[/tex]:
[tex]\[
-2(7) + 3(8) = -14 + 24 = 10 \quad (\text{which is } \geq 1)
\][/tex]
This point satisfies the first inequality.
For the second inequality, substitute [tex]\( x = 7 \)[/tex] and [tex]\( y = 8 \)[/tex]:
[tex]\[
-5(7) + 6(8) = -35 + 48 = 13 \quad (\text{which is not } \leq 1)
\][/tex]
This point does not satisfy the second inequality.
3. Point (8, 7):
For the first inequality, substitute [tex]\( x = 8 \)[/tex] and [tex]\( y = 7 \)[/tex]:
[tex]\[
-2(8) + 3(7) = -16 + 21 = 5 \quad (\text{which is } \geq 1)
\][/tex]
This point satisfies the first inequality.
For the second inequality, substitute [tex]\( x = 8 \)[/tex] and [tex]\( y = 7 \)[/tex]:
[tex]\[
-5(8) + 6(7) = -40 + 42 = 2 \quad (\text{which is not } \leq 1)
\][/tex]
This point does not satisfy the second inequality.
4. Point (9, 7):
For the first inequality, substitute [tex]\( x = 9 \)[/tex] and [tex]\( y = 7 \)[/tex]:
[tex]\[
-2(9) + 3(7) = -18 + 21 = 3 \quad (\text{which is } \geq 1)
\][/tex]
This point satisfies the first inequality.
For the second inequality, substitute [tex]\( x = 9 \)[/tex] and [tex]\( y = 7 \)[/tex]:
[tex]\[
-5(9) + 6(7) = -45 + 42 = -3 \quad (\text{which is } \leq 1)
\][/tex]
This point satisfies the second inequality.
The point [tex]\((9, 7)\)[/tex] satisfies both inequalities.
Therefore, the point [tex]\((9, 7)\)[/tex] is a solution to the system of inequalities.
