what-is-a-and-b-such-that-a-b-a-b-and-a-b-a-b-

Question Number 76531 by benjo 1/2 santuyy last updated on 28/Dec/19

w h a t i s a a n d b s u c h t h a t a + b = a \times b

a n d a + b = a / b ?

Commented by MJS last updated on 28/Dec/19

(1) a + b = a b \Rightarrow a = \frac{b}{b - 1} \Rightarrow b \neq 1

(2) \frac{b}{b - 1} + b = \frac{1}{b - 1}

\frac{b^{2}}{b - 1} = \frac{1}{b - 1}

b^{2} = 1 \land b \neq 1 \Rightarrow b = - 1 \Rightarrow a = \frac{1}{2}

Commented by john santu last updated on 28/Dec/19

t e s t i n g b = 1 t o e q u a t i o n \to a + 1 = a \times 1 t h a t r i g t, f o r a n y

v a l u e o f a

Commented by benjo 1/2 santuyy last updated on 28/Dec/19

b u t a \neq 0

Commented by MJS last updated on 28/Dec/19

b = 1 ? ? ?

a + 1 = a

no solution for a

Answered by john santu last updated on 28/Dec/19

\to a \times b = \frac{a}{b} [b \neq 0]

{a b}^{2} - a = 0 \to a = 0 \lor b = \pm 1

Commented by MJS last updated on 28/Dec/19

you forgot to check

a + b = \frac{a}{b} \Rightarrow b \neq 0

a + b = a b

a = 0 \Rightarrow b = 0

b = - 1 \Rightarrow a - 1 = - a \Rightarrow a = \frac{1}{2}

b = 1 \Rightarrow a + 1 = a \Rightarrow no solution for a

a + b = \frac{a}{b}

a = 0 \Rightarrow b = \frac{0}{b} \Rightarrow no solution for b

b = - 1 \Rightarrow a - 1 = - a \Rightarrow a = \frac{1}{2}

b = 1 \Rightarrow a + 1 = a \Rightarrow no solution for a

\Rightarrow

a = \frac{1}{2} \land b = - 1

Commented by john santu last updated on 28/Dec/19

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