. Let a, b, c, x, y be positive real numbers such that ax + by = bx + cy = cx + ay. Prove that b = c.

Question

This is a proof based algebra inequality problem, It has 5 variables and it's an inequality 

Answer 1

 

ax+by=bx+cy=cx+ay

ax+by=bx+cy and bx+cy=cx+ay

ax-bx=cy-by and bx-cx=ay-cy

(a-b)x=(c-b)y and (b-c)x=(a-c)y

x/y=(c-b)/(a-b) and y/x=(b-c)/(a-c)

x=c-b and y=a-b

x=a-c and y=b-c

c-b=a-c implies 2c=a+b

a-b=b-c implies 2b=a+c

2c-2b=a-a+b-c

2c-2b=b-c

2c+c=b+2b

3c=3b

c=b