2007 iTest Problems/Problem 22

Problem

Find the value of $c$ such that the system of equations $|x+y|=2007 \\ |x-y|=c$ has exactly two solutions $(x,y)$ in real numbers.

$\text{(A) } 0 \quad \text{(B) } 1 \quad \text{(C) } 2 \quad \text{(D) } 3 \quad \text{(E) } 4 \quad \text{(F) } 5 \quad \text{(G) } 6 \quad \text{(H) } 7 \quad \text{(I) } 8 \quad \text{(J) } 9 \quad \text{(K) } 10 \quad \text{(L) } 11 \quad \text{(M) } 12 \quad$

$\text{(N) } 13 \quad \text{(O) } 14 \quad \text{(P) } 15 \quad \text{(Q) } 16 \quad \text{(R) } 17 \quad \text{(S) } 18 \quad \text{(T) } 223 \quad \text{(U) } 678 \quad \text{(V) } 2007 \quad$

Solution

$2007|x-y|=c$ must be one line; otherwise, the system would have four solutions. This occurs at $c=\boxed{0\ \textbf{(A)}}$ .

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2007 iTest Problems/Problem 22

Problem

Solution