Answer:
b = 2, GH = 9
Step-by-step explanation:
The last two equations tell you ...
HI = 3 = 4b -5
8 = 4b . . . . . . . . . add 5
2 = b . . . . . . . . . . divide by 4
Then ...
GI = 5·2 +2 = 12
So ...
GI/HI = 12/3 = 4 = G/H . . . . use the found value for GI and the given HI
G = 4H . . . . . . . multiply by H
GH = 4H² . . . . . multiply by H again
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At this point, it can be useful to try the offered solutions to see what works.
For GH = 9
GH = 4H² = 9
H = ±√(9/4) = ±3/2
G = 4H = ±6
Then ...
I = 3/H = 3/(±3/2) = ±2 . . . . matching the sign of H
We want values of G and H such that H is between G and I.
positive H: (G, H, I) = (6, 3/2, 2/3) . . . . yes, H is between G and I
negative H: (G, H, I) = (-6, -3/2, -2/3) . . yes, H is between G and I
The solution b=2, GH=9 satisfies problem requirements.
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For GH = 12
GH = 4H² = 12
H = ±√(12/4) = ±√3
G = 4H = ±4√3
I = 3/H = 3/±√3 = ±√3 . . . . sign matches H
Then (G, H, I) = (4√3, √3, √3) . . . . . H = I; is not "between" G and I
or (G, H, I) = (-4√3, -√3, -√3) . . . . . . H = I; is not "between" G and I
The solution b=2, GH=12 will only satisfy the problem requirements if you interpret "between G and I" to allow that H = I.
