SOLUTION: Given that S=7a+b+7c and also S=6a+8b+6c, where 51<S<149 and a, b, c and S are integers, find the sum a+b+c. A) 14 B) 15 C) 16 D) 17 E) 18

Algebra ->  College  -> Linear Algebra -> SOLUTION: Given that S=7a+b+7c and also S=6a+8b+6c, where 51<S<149 and a, b, c and S are integers, find the sum a+b+c. A) 14 B) 15 C) 16 D) 17 E) 18      Log On


   

Question 1187121: Given that S=7a+b+7c and also S=6a+8b+6c, where 51
Answer by greenestamps(13198) About Me  (Show Source):
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S+=+7a%2Bb%2B7c [1]
S+=+6a%2B8b%2B6c [2]

Subtract [2] from [1]:

a-7b%2Bc+=+0
a%2Bc+=+7b
b+=+%28a%2Bc%29%2F7 [3]

Substitute [3] in [1]:



We know a, b, and c are integers, so S is an integer. From the last equation, we also know that S is a multiple of 50, and (a+c) is a multiple of 7.

We also know that S is between 51 and 149. The only multiple of 50 between 51 and 149 is 100. So

S+=+100+=+50%28%28a%2Bc%29%2F7%29
%28a%2Bc%29%2F7=2
a%2Bc=14 [4]

Substitute [4] in [3]:

b+=+14%2F2+=+7 [5]

Combine [4] and [5] to get the answer.

ANSWER: a+b+c = 14+2 = 16 -- answer choice C