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If x/y + y/x = 5, what is the value of (x + y)(1/x + 1/y)?
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28 Feb 2022, 04:00
28 Feb 2022, 04:00
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69% (01:09) correct
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If \(\frac{x}{y} + \frac{y}{x }= 5\), what is the value of \((x + y)(\frac{1}{x} + \frac{1}{y})\)?
(A) 3
(B) 7
(C) 10
(D) 15
(E) 25
(A) 3
(B) 7
(C) 10
(D) 15
(E) 25
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Posts: 6218
Given Kudos: 136
If x/y + y/x = 5, what is the value of (x + y)(1/x + 1/y)?
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28 Feb 2022, 05:45
28 Feb 2022, 05:45
1
Carcass wrote:
If \(\frac{x}{y} + \frac{y}{x }= 5\), what is the value of \((x + y)(\frac{1}{x} + \frac{1}{y})\)?
(A) 3
(B) 7
(C) 10
(D) 15
(E) 25
(A) 3
(B) 7
(C) 10
(D) 15
(E) 25
We want to find the value of: \((x + y)(\frac{1}{x} + \frac{1}{y})\)
Let's first use FOIL to expand the expression.
When we do so, we get: \( 1 + \frac{x}{y} + \frac{y}{x} + 1\)
Replace: \(\frac{x}{y} + \frac{y}{x }\) with \(5\) (from the given information) to get: \( 1 + 5 + 1\), which equals \(7\)
Answer: B
If x/y + y/x = 5, what is the value of (x + y)(1/x + 1/y)?
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22 Mar 2022, 22:16
22 Mar 2022, 22:16
1
We notice that in \((x + y)(\frac{1}{x} + \frac{1}{y})\) ,\( (\frac{1}{x}+\frac{1}{y})\) has the same denominator after cross-multiplication as \((\frac{x}{y} + \frac{y}{x})\) has.
Keeping this in mind, we will first cross multiply and add both the fractions in the brackets.
\((\frac{x}{y} + \frac{y}{x})\) = \(\frac{x^2 + y^2}{xy} \) = 5 (given)
\((\frac{1}{x} + \frac{1}{y})\) = \(\frac{x + y}{xy}\)
Now,
\((x + y)(\frac{1}{x} + \frac{1}{y})\) = \((x + y)\frac{x + y}{xy}\) =\( \frac{x^2 + y^2 + 2xy}{xy}\) = \(\frac{x^2 + y^2}{xy } + \frac{2xy}{xy}\) = \(5 + 2\) = \(7\)
Keeping this in mind, we will first cross multiply and add both the fractions in the brackets.
\((\frac{x}{y} + \frac{y}{x})\) = \(\frac{x^2 + y^2}{xy} \) = 5 (given)
\((\frac{1}{x} + \frac{1}{y})\) = \(\frac{x + y}{xy}\)
Now,
\((x + y)(\frac{1}{x} + \frac{1}{y})\) = \((x + y)\frac{x + y}{xy}\) =\( \frac{x^2 + y^2 + 2xy}{xy}\) = \(\frac{x^2 + y^2}{xy } + \frac{2xy}{xy}\) = \(5 + 2\) = \(7\)
