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Question Number 76531 by benjo 1/2 santuyy last updated on 28/Dec/19
what is a and b such that a+b = a×b and a+b =a/b ?
$${what}\:{is}\:{a}\:{and}\:{b}\:{such}\:{that}\:{a}+{b}\:=\:{a}×{b}\: \\ $$$${and}\:{a}+{b}\:={a}/{b}\:? \\ $$
Commented by MJS last updated on 28/Dec/19
(1) a+b=ab ⇒ a=(b/(b−1)) ⇒ b≠1 (2) (b/(b−1))+b=(1/(b−1)) (b^2 /(b−1))=(1/(b−1)) b^2 =1∧b≠1 ⇒ b=−1 ⇒ a=(1/2)
$$\left(\mathrm{1}\right)\:{a}+{b}={ab}\:\Rightarrow\:{a}=\frac{{b}}{{b}−\mathrm{1}}\:\Rightarrow\:{b}\neq\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:\frac{{b}}{{b}−\mathrm{1}}+{b}=\frac{\mathrm{1}}{{b}−\mathrm{1}} \\ $$$$\:\:\:\:\:\:\:\frac{{b}^{\mathrm{2}} }{{b}−\mathrm{1}}=\frac{\mathrm{1}}{{b}−\mathrm{1}} \\ $$$$\:\:\:\:\:\:\:{b}^{\mathrm{2}} =\mathrm{1}\wedge{b}\neq\mathrm{1}\:\Rightarrow\:{b}=−\mathrm{1}\:\Rightarrow\:{a}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$
Commented by john santu last updated on 28/Dec/19
testing b = 1 to equation → a + 1 =a×1 that rigt, for any value of a
$${testing}\:{b}\:=\:\mathrm{1}\:{to}\:{equation}\:\rightarrow\:{a}\:+\:\mathrm{1}\:={a}×\mathrm{1}\:{that}\:{rigt},\:{for}\:{any}\: \\ $$$${value}\:{of}\:{a} \\ $$
Commented by benjo 1/2 santuyy last updated on 28/Dec/19
but a ≠ 0
$${but}\:{a}\:\neq\:\mathrm{0} \\ $$
Commented by MJS last updated on 28/Dec/19
b=1 ??? a+1=a no solution for a
$${b}=\mathrm{1}\:??? \\ $$$${a}+\mathrm{1}={a} \\ $$$$\mathrm{no}\:\mathrm{solution}\:\mathrm{for}\:{a} \\ $$
Answered by john santu last updated on 28/Dec/19
→ a×b=(a/b) [b≠0] ab^2 −a=0 → a=0 ∨ b=±1
$$\rightarrow\:{a}×{b}=\frac{{a}}{{b}}\:\left[{b}\neq\mathrm{0}\right] \\ $$$${ab}^{\mathrm{2}} −{a}=\mathrm{0}\:\rightarrow\:{a}=\mathrm{0}\:\vee\:{b}=\pm\mathrm{1}\: \\ $$
Commented by MJS last updated on 28/Dec/19
you forgot to check a+b=(a/b) ⇒ b≠0 a+b=ab a=0 ⇒ b=0 b=−1 ⇒ a−1=−a ⇒ a=(1/2) b=1 ⇒ a+1=a ⇒ no solution for a a+b=(a/b) a=0 ⇒ b=(0/b) ⇒ no solution for b b=−1 ⇒ a−1=−a ⇒ a=(1/2) b=1 ⇒ a+1=a ⇒ no solution for a ⇒ a=(1/2)∧b=−1
$$\mathrm{you}\:\mathrm{forgot}\:\mathrm{to}\:\mathrm{check} \\ $$$${a}+{b}=\frac{{a}}{{b}}\:\Rightarrow\:{b}\neq\mathrm{0} \\ $$$${a}+{b}={ab} \\ $$$$\:\:\:\:\:{a}=\mathrm{0}\:\Rightarrow\:{b}=\mathrm{0} \\ $$$$\:\:\:\:\:{b}=−\mathrm{1}\:\Rightarrow\:{a}−\mathrm{1}=−{a}\:\Rightarrow\:{a}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\:\:\:\:\:{b}=\mathrm{1}\:\Rightarrow\:{a}+\mathrm{1}={a}\:\Rightarrow\:\mathrm{no}\:\mathrm{solution}\:\mathrm{for}\:{a} \\ $$$${a}+{b}=\frac{{a}}{{b}} \\ $$$$\:\:\:\:\:{a}=\mathrm{0}\:\Rightarrow\:{b}=\frac{\mathrm{0}}{{b}}\:\Rightarrow\:\mathrm{no}\:\mathrm{solution}\:\mathrm{for}\:{b} \\ $$$$\:\:\:\:\:{b}=−\mathrm{1}\:\Rightarrow\:{a}−\mathrm{1}=−{a}\:\Rightarrow\:{a}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\:\:\:\:\:{b}=\mathrm{1}\:\Rightarrow\:{a}+\mathrm{1}={a}\:\Rightarrow\:\mathrm{no}\:\mathrm{solution}\:\mathrm{for}\:{a} \\ $$$$\Rightarrow \\ $$$${a}=\frac{\mathrm{1}}{\mathrm{2}}\wedge{b}=−\mathrm{1} \\ $$
Commented by john santu last updated on 28/Dec/19
oo yes i agree
$${oo}\:{yes}\:{i}\:{agree} \\ $$

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