D01-43

Official Solution:


If \(5x = y + 7\), is \(x - y > 0\)?

(1) \(xy = 6\).

From the equation given in the stem, we know that \(x = \frac{y+7}{5}\). Substituting this into the original equation, we obtain \(\frac{y+7}{5}*y = 6\), which simplifies to \(y^2+7y-30=0\). Solving this quadratic equation gives \(y = -10\) or \(y=3\).

If \(y = -10\), then \(x = \frac{y+7}{5}=\frac{-3}{5}\), which means that in this case \(x > y\).

If \(y = 3\), then \(x = \frac{y+7}{5}=2\), which means that in this case \(x < y\).

Since we obtain different answers to the question of whether \(x > y\), statement (1) is insufficient.

(2) \(x\) and \(y\) are consecutive integers.

Note that \(x\) and \(y\) being consecutive integers does not necessarily imply that \(x < y\) and \(x + 1 = y\); it might be that \(x > y\) and \(x - 1 = y\).

If \(x - 1 = y\), then substituting into \(5x = y + 7\) gives \(5x = (x-1) + 7\), which gives \(x=\frac{3}{2}\) and \(y=\frac{1}{2}\). This is not valid since in the case \(x\) and \(y\) are not integers as given in the statement.

If \(x + 1 = y\), then substituting into \(5x = y + 7\) gives \(5x = (x+1) + 7\), which gives \(x=2\) and \(y=3\). Therefore, the answer to the question of whether \(x > y\) is NO. Statement (2) is sufficient.


Answer: B