Question : What are the values of $x$ and $y$, respectively, from the following equations?
$6x + 7y = 5xy$
$10y – 4x = 4xy$
Option 1: 3 and 4
Option 2: 4 and 5
Option 3: 2 and 4
Option 4: 2 and 5
Correct Answer: 2 and 4
Solution :
Here, we have,
$6x + 7y = 5xy$ .................( 1)
$10y - 4x = 4xy$.................( 2)
On dividing eq(1) and eq(2) by xy, we get,
$⇒ \frac{6}{y} + \frac{7}{x} = 5$............(3)
$⇒ \frac{10}{x} - \frac{4}{y} = 4$..............(4)
Now, multiply eq(3) by 2 and eq(4) by 3, we get,
$⇒ \frac{12}{y} + \frac{14}{x} = 10$ ...............(5)
$⇒ \frac{30}{x} - \frac{12}{y} = 12$..............( 6)
Now, eq(5) + eq(6)
$⇒ \frac{14}{x} + \frac{30}{x} = 22$
$⇒ \frac{44}{x} = 22$
$⇒x = 2 $
Now, put the value of $x = 2$ in the eq(1), we get,
$⇒ 6 × 2 + 7y = 5 × 2y$
$⇒ 12 = 3y$
$⇒ y = 4$
$\therefore$ The values of $x$ and $y$ are 2 and 4, respectively.
Hence, the correct answer is 2 and 4.
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