Given:
#a + b + c=15# ...................(1)
#c < b + a#...............................(2)
#b = (a + c)/2#..............................(3)
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Consider equation (3) #-> 2b=( a + c)#
Write equation (1) as
#( a+ c) + b=15#
By substitution this becomes
#2b+b=15#
#color(blue)(=> 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")#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#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")#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(green)("All determined values satisfy the given condition")#
#a=3" ; "b=5" ; "c=7#
