Solve-for-x-amp-y-1-x-1-y-5-1-x-2-1-y-2-13-

Question Number 217766 by ArshadS last updated on 20/Mar/25

S o l v e f o r x & y

\frac{1}{x} + \frac{1}{y} = 5

\frac{1}{x^{2}} + \frac{1}{y^{2}} = 13

Answered by Ghisom last updated on 20/Mar/25

2 + 3 = 5

4 + 9 = 13

x = \frac{1}{2} y = \frac{1}{3} \lor x = \frac{1}{3} y = \frac{1}{2}

Answered by nECxx2 last updated on 21/Mar/25

s o l u t i o n

f r o m (1)

\frac{x + y}{x y} = 5

f r o m (2)

\frac{x^{2} + y^{2}}{{(x y)}^{2}} = 13

\frac{{(x + y)}^{2} - 2 x y}{{(x y)}^{2}} = 13

∴ l e t a = x + y, b = x y

\frac{a}{b} = 5 (3)

\frac{a^{2} - 2 b}{b^{2}} = 13 (4)

{(\frac{a}{b})}^{2} - 2 (\frac{1}{b}) = 13

5^{2} - 2 / b = 13

\frac{2}{b} = 25 - 13

b = \frac{1}{6}

a = \frac{5}{6}

s u b s t i t u t e f o r x a n d y

x = \dots . y = \dots .

Commented by ArshadS last updated on 22/Mar/25

T h a n k s!

Answered by ArshadS last updated on 21/Mar/25

\frac{1}{x^{2}} + \frac{2}{x y} + \frac{1}{y^{2}} = 13 + \frac{2}{x y}

{(\frac{1}{x} + \frac{1}{y})}^{2} = 13 + \frac{2}{x y}

\frac{2}{x y} = 25 - 13 = 12

\frac{1}{x} \cdot \frac{1}{y} = 6

\frac{1}{x} (5 - \frac{1}{x}) = 6

\frac{5}{x} - {(\frac{1}{x})}^{2} = 6

{(\frac{1}{x})}^{2} - 5 (\frac{1}{x}) + 6 = 0

{(\frac{1}{x})}^{2} - 2 (\frac{1}{x}) - 3 (\frac{1}{x}) + 6 = 0

\frac{1}{x} (\frac{1}{x} - 2) - 3 (\frac{1}{x} - 2) = 0

(\frac{1}{x} - 2) (\frac{1}{x} - 3) = 0

x = \frac{1}{2} \lor x = \frac{1}{3}

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