Question

find x^2+y^2+z^2 if x^2+xy+xz=135,y^2+yz+yx=351 and z^2+zx+zy=243.

Answer 1

Dear Student,
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Please find below the solution to the asked query:

Given : x2 + xy + xz  = 135             --- ( 1 ) ,

y2 + yz + yx  = 351                        --- ( 2 )

And

z2 + zx + zy  = 243                        --- ( 3 )

Now we add equation 1 , 2 and 3 and get

x2 + y2  + z2  + 2 xy + 2 yz  + 2 zx  = 729 

( x + y + z )2  = 729                                 ( We know : ( x + y + z )2  = x2 + y2  + z2  + 2 xy + 2 yz  + 2 zx )

( x + y + z )2  = 272

x + y + z  = 27                                        --- ( 4 )

Now we take equation 1 : x2 + xy + xz  = 135

x ( x + y + z ) = 135  , Substitute value from equation 4 and get 

27 x  =  135

x = 5

And we take equation 2 : y2 + yz + yx  = 351

y ( x + y + z ) = 351  , Substitute value from equation 4 and get 

27 y  =  351

y = 13

And we take equation 3 : z2 + zx + zy  = 243

z ( x + y + z ) = 243  , Substitute value from equation 4 and get 

27 z  =  243

z = 9

Therefore,

Value of x2 + y2 +z2  = 52  + 132 + 92 =  25 + 169 + 81  = 275                                       ( Ans )

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Answer 2

Answer:

275

Step-by-step explanation:

Given : x2 + xy + xz  = 135             --- ( 1 ) ,

y2 + yz + yx  = 351                        --- ( 2 )

And

z2 + zx + zy  = 243                        --- ( 3 )

Now we add equation 1 , 2 and 3 and get

x2 + y2  + z2  + 2 xy + 2 yz  + 2 zx  = 729 

( x + y + z )2  = 729                                 ( We know : ( x + y + z )2  = x2 + y2  + z2  + 2 xy + 2 yz  + 2 zx )

( x + y + z )2  = 272

x + y + z  = 27                                        --- ( 4 )

Now we take equation 1 : x2 + xy + xz  = 135

x ( x + y + z ) = 135  , Substitute value from equation 4 and get 

27 x  =  135

x = 5

And we take equation 2 : y2 + yz + yx  = 351

y ( x + y + z ) = 351  , Substitute value from equation 4 and get 

27 y  =  351

y = 13

And we take equation 3 : z2 + zx + zy  = 243

z ( x + y + z ) = 243  , Substitute value from equation 4 and get 

27 z  =  243

z = 9

Therefore,

Value of x2 + y2 +z2  = 52  + 132 + 92 =  25 + 169 + 81  =275

Thanks

Aarohi