Respuesta :
The point [tex](-2,2)[/tex] is a solution of the system given by [tex]y<2x+8[/tex] and [tex]y \geq -x-3[/tex]
Given point: [tex](-2,2)[/tex]
Given systems:
- [tex]y>-2x+2[/tex] and [tex]y>x+5[/tex]
- [tex]y<x+2[/tex] and [tex]y>x-1[/tex]
- [tex]y<2x+8[/tex] and [tex]y \geq -x-3[/tex]
- [tex]y<2x+3[/tex] and [tex]y \geq -2x-5[/tex]
To find: The system to which the given point is a solution
If a point is a solution of a system, then the coordinates of the point satisfies all the equation(s) or inequation(s) of the system. So, we can substitute the x & y coordinates of the given point into the inequalities of each of the given systems and check if the inequalities are satisfied by the coordinates of the point.
(1) [tex]y>-2x+2[/tex] and [tex]y>x+5[/tex]
Substitute the coordinates of the point [tex](-2,2)[/tex] into the inequalities of the system.
Put [tex]x=-2,y=2[/tex] in [tex]y>-2x+2[/tex] to get,
[tex]2>-2(-2)+2[/tex]
[tex]2>4+2[/tex]
[tex]2>6[/tex]
The above inequality is clearly impossible and thus, the coordinates of the given point does not satisfy this inequality.
This implies that the given point is not a solution of this system.
(2) [tex]y<x+2[/tex] and [tex]y>x-1[/tex]
Substitute the coordinates of the point [tex](-2,2)[/tex] into the inequalities of the system.
Put [tex]x=-2,y=2[/tex] in [tex]y<x+2[/tex] to get,
[tex]2<-2+2[/tex]
[tex]2<0[/tex]
The above inequality is clearly impossible and thus, the coordinates of the given point does not satisfy this inequality.
This implies that the given point is not a solution of this system.
(3) [tex]y<2x+8[/tex] and [tex]y \geq -x-3[/tex]
Substitute the coordinates of the point [tex](-2,2)[/tex] into the inequalities of the system.
Put [tex]x=-2,y=2[/tex] in [tex]y<2x+8[/tex] to get,
[tex]2<2(-2)+8[/tex]
[tex]2<-4+8[/tex]
[tex]2<4[/tex]
This is a true inequality. Then, the given point satisfies the first inequality of the system.
We will now check if the point satisfies the second inequality of the system.
Put [tex]x=-2,y=2[/tex] in [tex]y \geq -x-3[/tex] to get,
[tex]2 \geq -(-2)-3[/tex]
[tex]2 \geq 2-3[/tex]
[tex]2 \geq -1[/tex]
This is also a true inequality. Then, the given point also satisfies the second inequality of the system.
Thus, the given point is a solution of this system.
(4) [tex]y<2x+3[/tex] and [tex]y \geq -2x-5[/tex]
Substitute the coordinates of the point [tex](-2,2)[/tex] into the inequalities of the system.
Put [tex]x=-2,y=2[/tex] in [tex]y<2x+3[/tex] to get,
[tex]2<2(-2)+3[/tex]
[tex]2<-4+3[/tex]
[tex]2<-1[/tex]
The above inequality is clearly impossible and thus, the coordinates of the given point does not satisfy this inequality.
This implies that the given point is not a solution of this system.
Thus, we can see that the coordinates of the given point [tex](-2,2)[/tex] satisfies the inequalities of the third system only.
Then, the point [tex](-2,2)[/tex] is a solution of the system given by [tex]y<2x+8[/tex] and [tex]y \geq -x-3[/tex].
Learn more about geometric solutions of system of linear inequalities here:
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