If x+x+y=10 and x+ y-y=12, find x+y.

1988 AMC 12 Problem 17

Attempt Problem 17 of the 1988 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1988 AMC 12 solutions, or check the answer key.

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17.

\lvert x\rvert+x+y=10

and

x+\lvert y\rvert-y=12,

find

x+y.

-2

2

\frac{18}{5}

\frac{22}{3}

22

Use each equation to rule out one sign possibility

After determining the signs of

x

and

y,

solve a linear system

x\le0,

the first equation gives

y=10,

contradicting the second; hence

x>0.

y\ge0,

the second gives

x=12,

contradicting the first; hence

y<0.

The equations become

2x+y=10

and

x-2y=12.

Thus

x=\frac{32}{5},\ y=-\frac{14}{5},

and

x+y=\frac{18}{5}.

Thus the correct answer is C.