Question

Solving with full explanation​

Answer 1

Answer :-

The value of given fraction is 1 .

Solution :-

Given

• x² + y² + z² = xy + yz + zx

To find :-

Value of (3x²+7y²+5z²)/(5x²y²+7x²y²+3z²+x²)

⭐ Firstly we will solve the given one .

→ x² + y² +z² = xy + yz + zx

→ x² +y² +z² -xy -yz -zx = 0

Multiplying and dividing by 2 .

→ ½ ( 2x² + 2y² + 2z² - 2xy - 2xz - 2yz ) = 0

→ ( x² - 2xy + y²)+(y²-2yz +z²)+(z²-2xz+x²) = 0

→ ( x-y)² + (y-z)² + (z-x)² = 0

The sum of three non negative term can zero only if they are zero . So,

→ x = y = z

Now replacing y and z by x

→ 3x⁴+5x⁴+7x⁴/5x²x²+7x²x²+3x²x²

→ 15x⁴/5x⁴+7x⁴+3x⁴

→ 15x⁴/15x⁴ = 1 .

Hence the answer of this algebraic expression is 1 .

Answer 2

=>x^2+y^2+z^2-xy-yz-zx=0

multiply the given equation by 2 and divivide by 2

=>1/2(2x^2+2y^2+2z^2-2xy-2yz-2xz)=0

on expanding this,we get

x^2+y^2-2xy+y^2+z^2-2yz+z^2+x^2-2zx=0

(x-y)^2+(y-z)^2+(z-x)^2=0

since all the three terms will be non-negative,so,they can only be zero if

x-y=0.....I)

y-z=0,.....ii)

z-x=0.......III)

from these three equation we can say

x=y=z

changing the given equation in terms of x ,we get

(3x^4+7x^4+5x^4)/(5x^4+7x^4+3x^4)

15x^4/15x^4=1

Hence the value will be 1