Show that the given proportions are equivalent.
a+c / b+d = a-c / d-b
and a / b = c / d
a/b=c/d
->ad=bc
a+c / b+d = a-c / b-d
->(a+c)(b-d)=(a-c)(b+d)
->ab+cb-ad-dc=ab+ad-cb-cd
->2ad=2bc
so the given proportions are equivalent
To show that the given proportions are equivalent, we need to prove that one proportion can be algebraically manipulated to obtain the other proportion.
Let's start with the first proportion:
(a+c) / (b+d) = (a-c) / (d-b)
To manipulate this proportion, we can cross-multiply:
(a+c) * (d-b) = (a-c) * (b+d)
Expanding both sides:
ad - ab + cd - bc = ab + ad - cb - cd
Rearranging the terms:
ad - ab - ad + cb - bc - cd = 0
We can now combine like terms:
cb - bc - cd + ad - ab - ad = 0
Simplifying further:
cb - bc - cd - ab = 0
Now, let's work with the second proportion:
a / b = c / d
To manipulate this proportion, we can cross-multiply:
a * d = b * c
Now, let's compare the two equations we have obtained:
cb - bc - cd - ab = 0 (Equation 1)
a * d = b * c (Equation 2)
Upon comparison, we can see that Equation 1 is identical to Equation 2. Therefore, we have successfully shown that the given proportions are equivalent.