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In the xy-coordinate plane, does the region bounded by \(x > 0\), \(y < 0\), and \(-y + 2x < 4\) contain the point \((a, \ b)\), where \(a\) and \(b\) are integers?
(1) \(ab = -4\)
(2) \(a + 4b = 0\)
(1) \(ab = -4\)
(2) \(a + 4b = 0\)
24 Sep 2021, 03:02
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Official Solution:
In the xy-coordinate plane, does the region bounded by \(x > 0\), \(y < 0\), and \(-y + 2x < 4\) contain the 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 at least 4, not less than 4. So, the question asks whether \(a = 1\) and \(b = -1\). (In other words, point \((1, \ -1)\) is the ONLY point that 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, 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, point \((a, \ b)\) is not in the region. Sufficient.
For better understanding, please see the image below:
The question asks whether the point \((a, \ b)\) is in the yellow region. As you can see, the only point in this region with integer coordinates is (-1, 1), so the question asks whether \(a = 1\) and \(b = -1\).
Answer: D
In the xy-coordinate plane, does the region bounded by \(x > 0\), \(y < 0\), and \(-y + 2x < 4\) contain the 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 at least 4, not less than 4. So, the question asks whether \(a = 1\) and \(b = -1\). (In other words, point \((1, \ -1)\) is the ONLY point that 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, 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, point \((a, \ b)\) is not in the region. Sufficient.
For better understanding, please see the image below:
The question asks whether the point \((a, \ b)\) is in the yellow region. As you can see, the only point in this region with integer coordinates is (-1, 1), so the question asks whether \(a = 1\) and \(b = -1\).
Answer: D
Blair15
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08 Jan 2024, 22:19
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Bunuel
For the first option, why should a be positive and b be negative? Question doesn't say that point lies on the line so a & b can take any values.
Where am I going wrong, pls help
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