Given:
a + b + c=15a+b+c=15 ...................(1)
c < b + ac<b+a...............................(2)
b = (a + c)/2b=a+c2..............................(3)
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Consider equation (3) -> 2b=( a + c)→2b=(a+c)
Write equation (1) as
( a+ c) + b=15(a+c)+b=15
By substitution this becomes
2b+b=152b+b=15
color(blue)(=> b=5)⇒b=5
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Now we have:
a + 5 + c=15 ...................(1_a)
c < 5 + a..............................(2_a)
5 = (a + c)/2..............................(3_a)
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From 1_a" " a + c=10 ->color(green)( a = 10 - c)
From 2_a" "c color(green)(- a) <5
thus c color(green)( - a) < 5" substitute for a "-> c color(green)( -(10-c)) < 5
=>2c < 15
color(blue)(=> c < 7 1/2)
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Suppose c=7 then equation (1) becomes
color(brown)(a + b + c=15color(blue)(" " ->" " a+5+7=15)
color(blue)("Thus, if c=7 then a = 3")
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color(blue)("Check using eqns (1) to (3)")
Equation 1" "->3+5+7 = 15" "color(red)("True")
Equation 2" "->7<5+3" "color(red)(" True")
Equation 3" "->5=(3+7)/2" "color(red)(" True")
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color(green)("All determined values satisfy the given condition")
a=3" ; "b=5" ; "c=7
