SOLUTION: if x=c*y+b*z y=a*z+c*x z=b*x+a*y Then prove that x^2/1-a^2=y^2/1-b^2=z^2/1-c^2

Algebra ->  Expressions -> SOLUTION: if x=c*y+b*z y=a*z+c*x z=b*x+a*y Then prove that x^2/1-a^2=y^2/1-b^2=z^2/1-c^2       Log On


   

Question 252008: if x=c*y+b*z
y=a*z+c*x
z=b*x+a*y
Then prove that x^2/1-a^2=y^2/1-b^2=z^2/1-c^2
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

if x=cy%2Bbz
y=az%2Bcx
z=bx%2Bay

Then prove that x%5E2%2F%281-a%5E2%29=y%5E2%2F%281-b%5E2%29=z%5E2%2F%281-c%5E2%29
                
Solve the second for c:
y=az%2Bcx
y-az=cx
%28y-az%29%2Fx=c
Use the third equation to substitute for z
%28y-a%28bx%2Bay%29%29%2Fx%7D=c%7D%7D%0D%0A%7B%7B%7B%28y-abx-a%5E2y%29%2Fx=c

Use this and the third original equation to substitute
for both c and z in the first equation:


x=cy%2Bbz
x=%28%28y-abx-a%5E2y%29%2Fx%29y%2Bb%28bx%2Bay%29
Multiply thru by x:
x%5E2=%28y-abx-a%5E2y%29y%2Bbx%28bx%2Bay%29
x%5E2=y%5E2-abxy-a%5E2y%5E2%2Bb%5E2x%5E2%2Babxy
x%5E2=y%5E2-cross%28abxy%29-a%5E2y%5E2%2Bb%5E2x%5E2%2Bcross%28abxy%29
x%5E2=y%5E2-a%5E2y%5E2%2Bb%5E2x%5E2
x%5E2-b%5E2x%5E2=y%5E2-a%5E2y%5E2
x%5E2%281-b%5E2%29=y%5E2%281-a%5E2%29

Divide both sides by %281-a%5E2%29%281-b%5E2%29





x%5E2%2F%281-a%5E2%29=y%5E2%2F%281-b%5E2%29 

To prove that one of these also equals z%5E2%2F%281-c%5E2%29
is exactly similar to the above.

Edwin