Answer:
(1,2)
Step-by-step explanation:
We have the system: [tex]\left \{ {{y=-x^2+x+2} \atop {y=-x+3}} \right.[/tex] to see which points represents the solution(s) of the system we have to replace them in both equations and see if the points verify them.
Point (1,2):
x=1, y=2
First equation:
[tex]y=-x^2+x+2\\2=-(1)^2+1+2\\2=-1+1+2\\2=2[/tex]
Is verified.
Second equation:
[tex]y=-x+3\\2=-1+3\\2=2[/tex]
Is verified.
Then this is the correct answer.
Points (1,2) and (0,3):
We saw that the point (1,2) verified both equations.
Now we are going to replace (0,3) in both equations:
[tex]y=-x^2+x+2\\3=-(0)^2+0+2\\3\neq2[/tex]
It isn't verified for the first equation.
[tex]y=-x+3\\3=-0+3\\3=3[/tex]
It is verified for the second.
Then this is not the solution of the system because (0,3) belongs only to the second equation.
Points (-1,0) and (2,0):
We have to replace (-1,0) in both equations:
First equation:
[tex]y=-x^2+x+2\\0=-(-1)^2-1+2\\0=-2+2\\0=0[/tex]
Is verified.
Second equation:
[tex]y=-x+3\\0=-1+3\\0\neq2[/tex]
It is not verified for the second equation. The point (-1,0) belongs to the graph of the first equation only.
We are going to replace (2,0) in the first equation:
[tex]y=-x^2+x+2\\0=-(2)^2+2+2\\0=-4+4\\0=0[/tex]
Is verified.
Now in the second equation:
[tex]y=-x+3\\0=-2+3\\0\neq1[/tex]
It isn't verified for the second equation. The point (2,0) belongs to the first equation.
Then this doesn't represent the solution.
Point (2,1):
First equation:
[tex]y=-x^2+x+2\\1=-(2)^2+2+2\\1=-4+4\\1\neq2[/tex]
It isn't verified for the first equation.
Second equation:
[tex]y=-x+3\\1=-2+3\\1=1[/tex]
Is verified.
Then this answer is incorrect, because the point is verified only for the second equation.
The graph of the system is attached, where [tex]y=-x^2+x+2[/tex] is the blue one and [tex]y=-x+3[/tex] the pink one.
