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20 Mar 2024, 02:52
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If \(x\) and \(y\) are integers such that \(4x + 3y = 3xy\), how many distinct pairs of \((x, y)\) are there?
A. 0
B. 1
C. 2
D. 3
E. 6
A. 0
B. 1
C. 2
D. 3
E. 6
20 Mar 2024, 02:52
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Official Solution:
If \(x\) and \(y\) are integers such that \(4x + 3y = 3xy\), how many distinct pairs of \((x, y)\) are there?
A. 0
B. 1
C. 2
D. 3
E. 6
There are several ways to solve this question through algebraic manipulations. Essentially, our goal is to arrive at an equation of the form \(x = expression \ with \ y \ only\) or \(y = expression \ with \ x \ only\).
\(4x + 3y = 3xy\)
\(3y = 3xy-4x\)
\(3y = x(3y-4)\)
\(x=\frac{3y}{3y-4}\)
Next, let's attempt to split the fraction to isolate \(y\) in a single term:
\(x=\frac{3y-4+4}{3y-4}\)
\(x=\frac{3y-4}{3y-4} + \frac{4}{3y-4}\)
\(x=1+ \frac{4}{3y-4}\)
Since \(x\) is an integer, for \(1+ \frac{4}{3y-4}\) to be an integer, \(\frac{4}{3y-4}\) must also be an integer. This means that \(3y-4\) must be a divisor of 4. The number 4 has six factors: -4, -2, -1, 1, 2, and 4. However, for -2, 1, and 4, \(y\) does not result in an integer. Therefore, \(3y-4\) can only be -4, -1, or 2. Accordingly, \(y\) can only be 0, 1, or 2. For each of these \(y\) values, we find a corresponding \(x\) value, leading to a total of three distinct pairs of \((x, y)\).
Answer: D
If \(x\) and \(y\) are integers such that \(4x + 3y = 3xy\), how many distinct pairs of \((x, y)\) are there?
A. 0
B. 1
C. 2
D. 3
E. 6
There are several ways to solve this question through algebraic manipulations. Essentially, our goal is to arrive at an equation of the form \(x = expression \ with \ y \ only\) or \(y = expression \ with \ x \ only\).
\(4x + 3y = 3xy\)
\(3y = 3xy-4x\)
\(3y = x(3y-4)\)
\(x=\frac{3y}{3y-4}\)
\(x=\frac{3y-4+4}{3y-4}\)
\(x=\frac{3y-4}{3y-4} + \frac{4}{3y-4}\)
\(x=1+ \frac{4}{3y-4}\)
Answer: D
23 Aug 2024, 23:41
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Bunuel
If we divide both sides by xy, we get-> (4/y) + (3/x) = 3<br />
We can have only 2 pairs of (x,y) which will yield to 3 i.e, (1,2) and (-3,1). If y cannot be equal to 0 as that would make 4/y as not defined.<br />
Let me know if I am incorrect.
