(a+bz)/(a+b)=(c+dz)/(d+c) if bc=ad

See below.

Step-by-step explanation:

(a+bz)/(a+b)=(c+dz)/(d+c) if bc=ad

bc = ad gives a = bc/d.

Substituting for a on the left side of the first equation:

(bc/d + bz)(bc/d + b)  =  (bc + bdz) / d  * d/ (bc + bd)

= (bc + bdz)(bc + bd)

But bc = ad so we have:

(ad + bdz) / (ad + bd)   (A)

From bc = ad we get c = ad/b.

Substituting for c on the right side of the first equation:

(ad/b + bz)(d + ad/b) =  (ad + bdz) / b  * b / (bd + ad)

= (ad + bdz ) / ( ad + bd) which is equal to (A) above.

So the original identity is true.

step-by-step explanation: