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Question Number 202616 by hardmath last updated on 30/Dec/23
If (a/(a + b)) − ((1 − a)/(a − b)) = x Find ((1 − b)/(a − b)) + (b/(a + b)) = ?
$$\mathrm{If}\:\:\:\:\:\frac{\mathrm{a}}{\mathrm{a}\:+\:\mathrm{b}}\:−\:\frac{\mathrm{1}\:−\:\mathrm{a}}{\mathrm{a}\:−\:\mathrm{b}}\:=\:\mathrm{x} \\ $$$$\mathrm{Find}\:\:\:\:\:\frac{\mathrm{1}\:−\:\mathrm{b}}{\mathrm{a}\:−\:\mathrm{b}}\:+\:\frac{\mathrm{b}}{\mathrm{a}\:+\:\mathrm{b}}\:=\:? \\ $$
Answered by MATHEMATICSAM last updated on 30/Dec/23
(a/(a + b)) − ((1 − a)/(a − b)) = x ⇒ (a/(a + b)) − 1 − ((1 − a)/(a − b)) − 1 = x − 2 ⇒ ((a − a − b)/(a + b)) − [((1 − a)/(a − b)) + 1] = x − 2 ⇒ − (b/(a + b)) − [((1 − a + a − b)/(a − b))] = x − 2 ⇒ − (b/(a + b)) − (((1 − b))/(a − b)) = x − 2 ⇒ ((1 − b)/(a − b)) + (b/(a + b)) = 2 − x
$$\frac{{a}}{{a}\:+\:{b}}\:−\:\frac{\mathrm{1}\:−\:{a}}{{a}\:−\:{b}}\:=\:{x} \\ $$$$\Rightarrow\:\frac{{a}}{{a}\:+\:{b}}\:−\:\mathrm{1}\:−\:\frac{\mathrm{1}\:−\:{a}}{{a}\:−\:\:{b}}\:−\:\mathrm{1}\:=\:{x}\:−\:\mathrm{2} \\ $$$$\Rightarrow\:\frac{{a}\:−\:{a}\:−\:{b}}{{a}\:+\:{b}}\:−\:\left[\frac{\mathrm{1}\:−\:{a}}{{a}\:−\:{b}}\:+\:\mathrm{1}\right]\:=\:{x}\:−\:\mathrm{2} \\ $$$$\Rightarrow\:−\:\frac{{b}}{{a}\:\:+\:{b}}\:−\:\left[\frac{\mathrm{1}\:−\:{a}\:+\:{a}\:−\:{b}}{{a}\:−\:{b}}\right]\:=\:{x}\:−\:\mathrm{2} \\ $$$$\Rightarrow\:−\:\frac{{b}}{{a}\:+\:{b}}\:−\:\frac{\left(\mathrm{1}\:−\:{b}\right)}{{a}\:−\:{b}}\:=\:{x}\:−\:\mathrm{2} \\ $$$$\Rightarrow\:\frac{\mathrm{1}\:−\:{b}}{{a}\:−\:{b}}\:+\:\frac{{b}}{{a}\:+\:{b}}\:=\:\mathrm{2}\:−\:{x} \\ $$
Answered by Rasheed.Sindhi last updated on 30/Dec/23
(a/(a + b)) − ((1 − a)/(a − b)) = x ((1 − b)/(a − b)) + (b/(a + b)) = y (say) x+y=((a+b)/(a+b))+((1−b−1+a)/(a−b))=1+1=2 y=2−x
$$\frac{\mathrm{a}}{\mathrm{a}\:+\:\mathrm{b}}\:−\:\frac{\mathrm{1}\:−\:\mathrm{a}}{\mathrm{a}\:−\:\mathrm{b}}\:=\:\mathrm{x} \\ $$$$\:\:\frac{\mathrm{1}\:−\:\mathrm{b}}{\mathrm{a}\:−\:\mathrm{b}}\:+\:\frac{\mathrm{b}}{\mathrm{a}\:+\:\mathrm{b}}\:=\:{y}\:\left({say}\right) \\ $$$${x}+{y}=\frac{{a}+{b}}{{a}+{b}}+\frac{\mathrm{1}−{b}−\mathrm{1}+{a}}{{a}−{b}}=\mathrm{1}+\mathrm{1}=\mathrm{2} \\ $$$${y}=\mathrm{2}−{x} \\ $$
Commented by hardmath last updated on 30/Dec/23
cool dear professors thankyou
$$\mathrm{cool}\:\mathrm{dear}\:\mathrm{professors}\:\mathrm{thankyou} \\ $$

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