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5(X^2+Y^2+Z^2)=4(XY+YZ+ZX) then find X:Y:Z

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The answer should be -X=Y=Z=0.
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3 XYZ
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IITIAN DHARMADAS (M.Sc. from IIT GUWAHATI)

Solving the equation you will get (2x-y)^2+(2y-z)^2+(2z-x)^2=0 Since sum of three square number is zero so they are individually equal to 0. So 2x-y=0; 2y-z=0; 2z-x=0 From these three equation we get x=y=z=0
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MATHEMATICS

1:1:1
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Computer Professional Teacher with more than 15 years of experience in Computer Science

X:Y:Z=1:1:1
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0:0:0
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A Teacher with 20 years of experience.

In this condition, there is no exact value given for the equation. Hence we cannot solve it without a given state. We cannot find a specific answer for each variable as there are no values given. I also tried the trial and error method, but the answer keeps fluctuating. We might have to create assumptions,... read more

In this condition, there is no exact value given for the equation. Hence we cannot solve it without a given state. We cannot find a specific answer for each variable as there are no values given. I also tried the trial and error method, but the answer keeps fluctuating. We might have to create assumptions, and then we can solve it.

read less
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Tutor with an experience in teaching Mathematics since 6 years.

5(x2 + y2 + z2) = 4(xy + yz + zx) or, 5x2 + 5y2 + 5z2 = 4xy + 4yz + 4zx or, 4x2 + 4y2 + 4z2 + x2 + y2 + z2 = 4xy + 4yz + 4zx or, (4x2 – 4xy + y2) + (4y2 – 4yz + z2) + (4z2 – 4zx + x2) = 0 or, (2x - y)2 + (2y - z)2 + (2z - x)2 = 0 We know that, since sum of squares is 0, each square... read more

5(x2 + y2 + z2) = 4(xy + yz + zx)

or, 5x2 + 5y2 + 5z2 = 4xy + 4yz + 4zx

or, 4x2 + 4y2 + 4z2 + x2 + y2 + z2 = 4xy + 4yz + 4zx

or, (4x2 – 4xy + y2) + (4y2 – 4yz + z2) + (4z2 – 4zx + x2) = 0

or, (2x - y)2 + (2y - z)2 + (2z - x)2 = 0

We know that, since sum of squares is 0, each square must be equal to 0.

Therefore,
                   (2x - y)2 = (2y - z)2 = (2z - x)2 = 0

             or, (2x - y) = (2y - z) = (2z - x) = 0

Therefore,

                       2x = y, 2y = z, 2z = x.

Thus converting all terms in respect to ‘x’ we get,
                                           y = 2x
                                           z = x/2
Therefore,
                   x : y : z = x : 2x : x/2
                                = 1 : 2 : ½
                                = 2 : 4 : 1

Answer:      x : y : z = 2 : 4 : 1

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Tutor with an experience in teaching Mathematics since 6 years.

5(x2 + y2 + z2) = 4(xy + yz + zx) or, 5x2 + 5y2 + 5z2 = 4xy + 4yz + 4zx or, 4x2 + 4y2 + 4z2 + x2 + y2 + z2 = 4xy + 4yz + 4zx or, (4x2 – 4xy + y2) + (4y2 – 4yz + z2) + (4z2 – 4zx + x2) = 0 or, (2x - y)2 + (2y - z)2 + (2z - x)2 = 0 We know that, since sum of squares is 0, each square... read more

5(x2 + y2 + z2) = 4(xy + yz + zx)

or, 5x2 + 5y2 + 5z2 = 4xy + 4yz + 4zx

or, 4x2 + 4y2 + 4z2 + x2 + y2 + z2 = 4xy + 4yz + 4zx

or, (4x2 – 4xy + y2) + (4y2 – 4yz + z2) + (4z2 – 4zx + x2) = 0

or, (2x - y)2 + (2y - z)2 + (2z - x)2 = 0

We know that, since sum of squares is 0, each square must be equal to 0.

Therefore,
                   (2x - y)2 = (2y - z)2 = (2z - x)2 = 0

             or, (2x - y) = (2y - z) = (2z - x) = 0

Therefore,

                       2x = y, 2y = z, 2z = x.

Thus converting all terms in respect to ‘x’ we get,
                                           y = 2x
                                           z = x/2
Therefore,
                   x : y : z = x : 2x : x/2
                                = 1 : 2 : ½
                                = 2 : 4 : 1

Answer:      x : y : z = 2 : 4 : 1

read less
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1:1:1
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