Question

A+B=AXB=A÷B What are the values of A & B?​

Answer 1

A = 1/2  and B = - 1

Given:

A+B=A×B=A/B

To Find:

The values of A and B.

Solution:

We have,

A + B = A × B = A /B  → [1]

Let us assume that,

A + B = A × B = A /B = k

Therefore,

From A/B = k,

We have,

A= Bk → [2]

Substituting A by Bk on [1],

We have,

Bk + B = Bk × B  = Bk/B

or, B (k + 1)  = B²k  =  k → [3]

Now, again, from [3],

B²k   =  k

or, B² = 1

or, B = ± 1

And,

B (k + 1)  = B²k  

or, k + 1 = Bk → [4]

Now, by using B = 1 in [4]

k + 1 = k

or, 0 = 1 , which is not possible

So, now by using B = - 1 in [4]

=> k + 1  = - k

=> 2k = -1

=>  k = -1/2

Thus,

When B = -1  ,

k =  -1/2  ,

Now to find A,

From [2],

A = Bk

or,A =  -1(-1/2) [Putting the value of k and B]

or, A = 1/2

Thus, we get the values of A and B as,

A = 1/2

B = - 1

Now, let us check if the equation is getting satisfied.

so,

A + B = 1/2 - 1 = -1/2

A × B = (1/2)(-1) = -1/2

A ÷ B (1/2) ÷ (-1) = - 1/2

i.e, A+B=A × B=A/B = [Proved]

Hence A = 1/2  and B = - 1.

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