Answer:
1
Explanation:
Label the points as in the attachment. Then we have ...
- P = (a+b)/2
- Q = (b+c)/2
- R = (c+d)/2
- S = (d+e)/2
- T = (e+a)/2
We can form the sum P + R + T and we get ...
P +R +T = (a+b)/2 +(c+d)/2 +(e+a)/2 = a +(b +c +d +e)/2
We can form the sum Q + S and we get ...
Q + S = (b+c)/2 +(d+e)/2 = (b +c +d +e)/2
Subtracting the latter sum from the former one gives ...
P +R +T -(Q +S) = a +(b +c +d +e)/2 -(b +c +d +e)/2 = a
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So, the value picked by the person with the average "6" was ...
(7 +1 +5) -(9 +3) = 13 -12 = 1
The person with average "6" picked 1.
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The system of equations written in matrix form is shown in the second attachment. The inverse of the coefficient matrix is shown in the third attachment. That is where the sum shown above came from.
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The rest of the picked numbers are ...
P = 2, b = 13, Q = 14, c = 5, R = 6, d = -3, S = -2, e = 9, T = 10