To determine whether the point [tex]\((6, -1)\)[/tex] is a solution to the given system of inequalities, we will substitute the coordinates of the point into each inequality and check if both inequalities are satisfied.
The system of inequalities is:
[tex]\[ 11x - 7y \geq 56 \][/tex]
[tex]\[ x + 7y < 28 \][/tex]
Let's substitute [tex]\( x = 6 \)[/tex] and [tex]\( y = -1 \)[/tex] into both inequalities:
### Checking the first inequality:
[tex]\[ 11x - 7y \geq 56 \][/tex]
Substitute [tex]\( x = 6 \)[/tex] and [tex]\( y = -1 \)[/tex]:
[tex]\[ 11(6) - 7(-1) \geq 56 \][/tex]
[tex]\[ 66 + 7 \geq 56 \][/tex]
[tex]\[ 73 \geq 56 \][/tex]
This inequality is true because [tex]\( 73 \geq 56 \)[/tex].
### Checking the second inequality:
[tex]\[ x + 7y < 28 \][/tex]
Substitute [tex]\( x = 6 \)[/tex] and [tex]\( y = -1 \)[/tex]:
[tex]\[ 6 + 7(-1) < 28 \][/tex]
[tex]\[ 6 - 7 < 28 \][/tex]
[tex]\[ -1 < 28 \][/tex]
This inequality is also true because [tex]\( -1 < 28 \)[/tex].
Since both inequalities are satisfied when the point [tex]\((6, -1)\)[/tex] is substituted into them, the point [tex]\((6, -1)\)[/tex] is indeed a solution to the system of inequalities.
Thus, the point [tex]\((6, -1)\)[/tex] is a solution to this system of inequalities.
