Bunuel wrote:
In the xy-coordinate plane, does region bounded by x > 0, y < 0, and -y + 2x < 4 contain point (a, b), where a and b are integers ?
(1) ab = -4
(2) a + 4b = 0
Official Solution:In the xy-coordinate plane, does region bounded by \(x > 0\), \(y < 0\), and \(-y + 2x < 4\) contain point \((a, \ b)\), where a and b are integers ? Let's check which set(s) of
integers \((a, \ b)\), where \(a > 0\) and \(b < 0\), satisfy \(-b + 2a < 4\). Notice that since \(b\) must be negative, then \(-b\) will be positive, so \(-b + 2a =(positive \ integer) + (even \ positive \ integer)\), which is at least 3. \(-b + 2a\) is 3, when \(b=-1\) and \(a=1\). For all other values, \(-b + 2a\) will be \( \geq 4\), not \(< 4\). So, the question basically asks whether \(a=1\) and \(b=-1\). (In other words, point \((1, \ -1)\) is the ONLY point which satisfies \(a > 0\), \(b < 0\), and \(-b + 2a < 4\).)
(1) \(ab = -4\). Since \(a=1\) and \(b=-1\) is not a solution of this equation, then point \((a, \ b)\) is not in the region. Sufficient.
(2) \(a + 4b = 0\). Since \(a=1\) and \(b=-1\) is not a solution of this equation, then point \((a, \ b)\) is not in the region. Sufficient.
For better understanding check the image below:
The question asks whether point \((a, \ b)\) is in the yellow region. As you can see the only point in this region which has integer coordinates is point (-1, 1), so the question aks whether \(a=1\) and \(b=-1\)
Answer: D