Once we have reciprocal we can define the division of xy of

Once we have reciprocal, we can define the division of x/y of two real numbers x, y by x/y:= x times y^-1, provided that y is non-zero. Prove that if x, y, z are real number such that xz = yx, then x =y, provided that z notequalto 0.

Solution

Given that division says that, x/y = x* y^-1 where y not equal to zero.

Given also x,y,z R(Real numbers) , where z is not equal to zero, and

xz = yz

for each x R – {0} , there exists a real number denoted by x^-1, such that    

x*x^-1 = 1

so we can write given xz = yz as following.

xz = yz

multiply both sides with z^-1 ,

z^-1(xz) = z^-1(yz)

(z^-1z)x = (z^-1z)y

(zz^-1)x = (zz^-1)y

1* x     = 1 * y

x = y.

hence proved as x = y   if xz = yz provided that z is not equal to zero.