Question

if x+y=25 and xcube + ycube = 4825 then what is the value of x and y

Answer 1

$Given \: x + y = 25 \: ---(1)$

$and \: x^{3} + y^{3} = 4825 \: --(2)$

$\implies (x+y)^{3} - 3xy(x+y) = 4825$

$\implies (x+y) [ (x+y)^{2} - 3xy] = 25 \times 193$

$\implies 25\times [ 25^{2} - 3xy] = 25 \times 193 \: [From \: (1) ]$

/* On Dividing bothsides by 25, we get */

$\implies 625 - 3xy = 193$

$\implies - 3xy = 193 - 625$

$\implies - 3xy = - 432$

/* On Dividing bothsides by (-3), we get */

$\implies xy = 144 \: ---(3)$

$Now, (x-y)^{2} = (x+y)^{2} - 4xy \\= 25^{2} - 4 \times 144 \: [ From \: (1) \:and \:(3) ] \\= 625 - 576 \\= 49$

$\implies x - y = 7 \: --(4)$

/* Adding equations (1) and (4) , we get */

$2x = 32$

$\implies x = \frac{32}{2}$

$\implies x = 16$

/* Put x = 16 in equation (1) , we get */

$16 + y = 24$

$\implies y = 24 - 16 = 8$

Therefore.,

$\green { x = 16 \: and \: y = 8 }$

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