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) y-15=3-15=-12 -x^2+4x=-(-2)^2+4*(-2)=-12 good S2:(7,-6) y-15=-6-15=-21 -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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