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2007 iTest Problems/Problem 2

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Problem

Find a + b if a and b satisfy 3a + 7b = 1977 and 5a + b = 2007.

\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498

Solution

3a + 7b = 1977 and 5a + b = 2007.

Thus, b=2007-5a, and substituting, 3a+14049-35a=1977\Rightarrow-32a=-12072\Rightarrow a=377.25. Thus, b=2007-1886.25\Rightarrow b=120.75. Thus, a+b=377.25+120.75=498\Rightarrow \boxed{\mathrm{B}}

Alternate Solution

We have 3a+7b=1977 and 5a+b=2007. Notice the symmetry we have if we add the two equations together: 8a+8b=3984. Dividing by 8, we have a+b=498. \boxed{\mathrm{B}}

See Also

2007 iTest (Problems)
Preceded by:
Problem 1 		Followed by:
Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60
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