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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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Maths Tutor

10
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Experienced (10+ Y) & Certified Mathematics Assistant Professor from Uttaranchal University, Dehradun

X=Y=Z
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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,...
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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...
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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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