2007 iTest Problems/Problem 2

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 $a+b=498$ . $\boxed{\mathrm{B}}$

2007 iTest (Problems, Answer Key)
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2007 iTest Problems/Problem 2

Contents

Problem

Solution

Alternate Solution

See Also