x-y-3-and-x-5-y-3-5-find-x-and-y-

Question Number 139331 by 7770 last updated on 26/Apr/21

\sqrt{x} + \sqrt{y} = 3 a n d

\sqrt{x + 5} + \sqrt{y + 3} = 5

f i n d x a n d y

Answered by bemath last updated on 26/Apr/21

(1) x = 9 + y - 6 \sqrt{y}

(2) x + 5 = 25 + y + 3 - 10 \sqrt{y + 3}

\Rightarrow 14 + y - 6 \sqrt{y} = 28 + y - 10 \sqrt{y + 3}

\Rightarrow 14 - 6 \sqrt{y} = 28 - 10 \sqrt{y + 3}

\Rightarrow 6 \sqrt{y} = 10 \sqrt{y + 3} - 14

\Rightarrow 3 \sqrt{y} = 5 \sqrt{y + 3} - 7

\Rightarrow 9 y + 42 \sqrt{y} + 49 = 25 (y + 3)

42 \sqrt{y} = 16 y + 26

\Rightarrow 21 \sqrt{y} = 8 y + 13, \sqrt{y} = u

\Rightarrow {8 u}^{2} - 21 u + 13 = 0

u = \frac{21 + \sqrt{441 - 32.13}}{16} = \frac{21 + 5}{16}

u = \frac{26}{16} \Rightarrow y = {(\frac{26}{16})}^{2} = \frac{676}{256}

Answered by MJS_new last updated on 26/Apr/21

x ⩾ 0 \land y ⩾ 0

(1) y = {(3 - \sqrt{x})}^{2}

(2) y = {(5 - \sqrt{x + 5})}^{2} - 3

\Rightarrow

{(3 - \sqrt{x})}^{2} - {(5 - \sqrt{x + 5})}^{2} + 3 = 0

5 \sqrt{x + 5} - 3 \sqrt{x} = 9

squaring and transforming

17 x + 22 = 15 \sqrt{x} \sqrt{x + 5}

squaring and transforming

x^{2} - \frac{377}{64} x + \frac{121}{16} = 0

x_{1} = 4 \land x_{2} = \frac{121}{64} \Rightarrow y_{1} = 1 \land y_{2} = \frac{169}{64}

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