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If \(xy ≠ 0\) and \(5x - 2y = 80\), how many pairs of \((x, y)\) exist, where \(x\) and \(y\) are integers with opposite signs?
A. 4
B. 5
C. 6
D. 7
E. 8
A. 4
B. 5
C. 6
D. 7
E. 8
08 Jan 2025, 00:19
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Official Solution:
If \(xy ≠ 0\) and \(5x - 2y = 80\), how many pairs of \((x, y)\) exist, where \(x\) and \(y\) are integers with opposite signs?
A. 4
B. 5
C. 6
D. 7
E. 8
If \(x\) is negative and \(y\) is positive, \(5x - 2y = negative - positive = negative\), which cannot equal the positive number 80. Thus, we are looking for pairs where \(x\) is positive and \(y\) is negative.
Rearranging, we get \(5x = 80 + 2y\).
Since the left-hand side is positive, \(y\) cannot be \(-40\) or less because that would make \(80 + 2y\) negative. Therefore, \(y\) must be more than -40 and less than 0.
Since the left-hand side is a multiple of 5, \(y\) must also be a multiple of 5 to ensure that the sum of 80 and \(2y\) is a multiple of 5. There are only seven multiples of 5 between -40 and 0: -35, -30, -25, -20, -15, -10, and -5.
Thus, there are seven pairs of \((x, y)\) possible.
Answer: D
If \(xy ≠ 0\) and \(5x - 2y = 80\), how many pairs of \((x, y)\) exist, where \(x\) and \(y\) are integers with opposite signs?
A. 4
B. 5
C. 6
D. 7
E. 8
If \(x\) is negative and \(y\) is positive, \(5x - 2y = negative - positive = negative\), which cannot equal the positive number 80. Thus, we are looking for pairs where \(x\) is positive and \(y\) is negative.
Rearranging, we get \(5x = 80 + 2y\).
Since the left-hand side is positive, \(y\) cannot be \(-40\) or less because that would make \(80 + 2y\) negative. Therefore, \(y\) must be more than -40 and less than 0.
Since the left-hand side is a multiple of 5, \(y\) must also be a multiple of 5 to ensure that the sum of 80 and \(2y\) is a multiple of 5. There are only seven multiples of 5 between -40 and 0: -35, -30, -25, -20, -15, -10, and -5.
Thus, there are seven pairs of \((x, y)\) possible.
Answer: D
17 Feb 2025, 12:38
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An alternative solution:
We can think about the expression 5x - 2y = 80 as the equation of a line with a y-intersection in (0,-40), an x-intersection in (16,0), and with a positive slope (a=+5).
Therefore, it is clear that the pairs of integers with opposite signs must be within the 4th quadrant. Since x and y must be different from zero, the set of integers pairs with opposite signs exists when x >0 and x <16. Rewriting the equation as 2y= 5x - 80, 2y must be a pair number, therefore x can assume only pair values from 2 to 14 (inclusive), which yields a total of 7 (x,y) pairs.
We can think about the expression 5x - 2y = 80 as the equation of a line with a y-intersection in (0,-40), an x-intersection in (16,0), and with a positive slope (a=+5).
Therefore, it is clear that the pairs of integers with opposite signs must be within the 4th quadrant. Since x and y must be different from zero, the set of integers pairs with opposite signs exists when x >0 and x <16. Rewriting the equation as 2y= 5x - 80, 2y must be a pair number, therefore x can assume only pair values from 2 to 14 (inclusive), which yields a total of 7 (x,y) pairs.
