Accepted Solution

An interesting question!

Instead of solving each set of non-linear system of equations 6 times, I will try a simpler way, which is more adapted to multiple choice questions.  However, please be warned that this method will not improve immensely math skills, but will help with reasoning, and possibly a broader understand in system of equations.

For simplicity, I will denote the sets of system of non-linear equations by S={A,B,C,D,E,F} from left to right.

1. given solutions {(-2,3),(7,-6)}
We first check each member of S, i.e. A,B,C,D,E,F for the linear conditions.
A: x+y=-2+3=1  ≠ 3   so no
B: x-y=-2-3=-5  ≠ 1   so no
C: 2x+y=-2(2)+3=-1  ≠ 1   so no
D: x+2y=-2+3(2)=4  ≠ 2   so no
E: -x+y=2+3=5  ≠ 1   so no
F: x+y=-2+3=1  ≠ 3   so YES now check the other solution 7-6=-1   OK

Now check the non-linear condition for F:
S1: (-2,3)
-x^2+4x=-(-2)^2+4*(-2)=-12   good
-x^2+4x=-(7^2)+4(7)=-49+28=-21 also good, 
So Solution set (1) matches tile F

(2) Given solutions {(-5,8),(3,0)}
We proceed in a similar way, to find that
-5+8=3 & 3+0=3 (matches A) and nothing else.
Check non-linear conditions (optionally, I transposed the terms to make comparison easier)
A. x^2+x-y=(-5)^2+(-5)-8=25-5-8=12   (looks good)
    x^2+x-y=(3)^2+(3)-0=9+3=12          (looks even better)
So we determined that {(-5,8),(3,0)} is the solution for tile A.

(3) For solution set {(-2,5),(3,-5)}
Check linear conditions:
x+y=3 and -3, so does not satisfy A,& F.
x-y=-7 and 8, so does not satisfy B
2x+y=5 and 1 so satisfies C
x+2y=8 & -7 so does not satisfy D
-x+y=7 & -8 so does not satisfy E
Thus C is our only set.  NOTE: if two distinct points satisfy one linear condition (say C), then both points cannot satisfy another non-equivalent linear condition, i.e. we didn't really have to check points D & E (which are not equivalent linear conditions as C) once we have found C.
Now check non-linear conditions:
x^2-3x-y=(-2)^2-3(-2)-5=5 ok
x^2-3x-y=(3)^2-3(3)-(-5)=5 ok
So solution set (3) matches tile C.

@officiallyqueenz, for your benefit, I will leave one for you as exercise.  Please proceed to find the system for solution set (4).  Post a your work as comments if you get stuck, or your answer for verification if you wish.

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