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Question Number 176037 by Shrinava last updated on 11/Sep/22
If (a−b)(a + b) = 13 Find 2a + b = ?
$$\mathrm{If}\:\:\:\left(\mathrm{a}−\mathrm{b}\right)\left(\mathrm{a}\:+\:\mathrm{b}\right)\:=\:\mathrm{13} \\ $$$$\mathrm{Find}\:\:\:\mathrm{2a}\:+\:\mathrm{b}\:=\:? \\ $$
Commented by Rasheed.Sindhi last updated on 11/Sep/22
20
$$\mathrm{20} \\ $$
Answered by Rasheed.Sindhi last updated on 11/Sep/22
Assuming a,b∈Z (a−b)(a + b) =1× 13=−1×−13 { ((a−b=1)),((a+b=13)) :}⇒a=7,b=6 2a+b=2(7)+6=20 OR { ((a−b=13)),((a+b=1)) :}⇒a=7,b=−6 2a+b=2(7)+(−6)=8 (a−b)(a + b) =−1×−13 { ((a−b=−1)),((a+b=−13)) :}⇒a=−7,b=−6 2a+b=2(−7)+(−6)=−20 OR { ((a−b=−13)),((a+b=−1)) :}⇒a=−7,b=6 2a+b=2(−7)+6=−8
$$\mathrm{Assuming}\:\mathrm{a},\mathrm{b}\in\mathbb{Z} \\ $$$$\left(\mathrm{a}−\mathrm{b}\right)\left(\mathrm{a}\:+\:\mathrm{b}\right)\:=\mathrm{1}×\:\mathrm{13}=−\mathrm{1}×−\mathrm{13} \\ $$$$\begin{cases}{\mathrm{a}−\mathrm{b}=\mathrm{1}}\\{\mathrm{a}+\mathrm{b}=\mathrm{13}}\end{cases}\Rightarrow\mathrm{a}=\mathrm{7},\mathrm{b}=\mathrm{6} \\ $$$$\mathrm{2a}+\mathrm{b}=\mathrm{2}\left(\mathrm{7}\right)+\mathrm{6}=\mathrm{20} \\ $$$$\mathrm{OR} \\ $$$$\begin{cases}{\mathrm{a}−\mathrm{b}=\mathrm{13}}\\{\mathrm{a}+\mathrm{b}=\mathrm{1}}\end{cases}\Rightarrow\mathrm{a}=\mathrm{7},\mathrm{b}=−\mathrm{6} \\ $$$$\mathrm{2a}+\mathrm{b}=\mathrm{2}\left(\mathrm{7}\right)+\left(−\mathrm{6}\right)=\mathrm{8} \\ $$$$ \\ $$$$\left(\mathrm{a}−\mathrm{b}\right)\left(\mathrm{a}\:+\:\mathrm{b}\right)\:=−\mathrm{1}×−\mathrm{13} \\ $$$$\begin{cases}{\mathrm{a}−\mathrm{b}=−\mathrm{1}}\\{\mathrm{a}+\mathrm{b}=−\mathrm{13}}\end{cases}\Rightarrow\mathrm{a}=−\mathrm{7},\mathrm{b}=−\mathrm{6} \\ $$$$\mathrm{2a}+\mathrm{b}=\mathrm{2}\left(−\mathrm{7}\right)+\left(−\mathrm{6}\right)=−\mathrm{20} \\ $$$$\mathrm{OR} \\ $$$$\begin{cases}{\mathrm{a}−\mathrm{b}=−\mathrm{13}}\\{\mathrm{a}+\mathrm{b}=−\mathrm{1}}\end{cases}\Rightarrow\mathrm{a}=−\mathrm{7},\mathrm{b}=\mathrm{6} \\ $$$$\mathrm{2a}+\mathrm{b}=\mathrm{2}\left(−\mathrm{7}\right)+\mathrm{6}=−\mathrm{8} \\ $$$$ \\ $$
Commented by Shrinava last updated on 11/Sep/22
a,b∈N
$$\mathrm{a},\mathrm{b}\in\mathbb{N} \\ $$
Commented by Rasheed.Sindhi last updated on 11/Sep/22
If a,b∈N: 2a+b=2(7)+6=20
$$\mathrm{If}\:\mathrm{a},\mathrm{b}\in\mathbb{N}: \\ $$$$\mathrm{2a}+\mathrm{b}=\mathrm{2}\left(\mathrm{7}\right)+\mathrm{6}=\mathrm{20} \\ $$
Commented by Shrinava last updated on 11/Sep/22
cool thank you dear ser
$$\mathrm{cool}\:\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{ser} \\ $$
Answered by BaliramKumar last updated on 11/Sep/22
a, b ∈ N (a−b)(a+b) = 1×13 a−b = 1 a+b = 13 a = 7, b = 6 2a+b = 2∙7 + 6 = 14 + 6 = 20 Answer ••▶ a, b ∈ Z (a, b) = (7, 6), (−7, −6), (7, −6), (−7, 6) 2a + b = 20, −20, 8, −8 Answers ••▶ a, b ∈ R ∞ solutions
$${a},\:{b}\:\in\:\mathbb{N} \\ $$$$\left({a}−{b}\right)\left({a}+{b}\right)\:=\:\mathrm{1}×\mathrm{13} \\ $$$${a}−{b}\:=\:\mathrm{1} \\ $$$${a}+{b}\:=\:\mathrm{13} \\ $$$${a}\:=\:\mathrm{7},\:\:\:\:\:{b}\:=\:\mathrm{6} \\ $$$$\mathrm{2}{a}+{b}\:=\:\mathrm{2}\centerdot\mathrm{7}\:+\:\mathrm{6}\:=\:\mathrm{14}\:+\:\mathrm{6}\:=\:\mathrm{20}\:\mathrm{Answer} \\ $$$$ \\ $$$$\bullet\bullet\blacktriangleright\:{a},\:{b}\:\in\:\mathbb{Z} \\ $$$$\left({a},\:{b}\right)\:=\:\left(\mathrm{7},\:\mathrm{6}\right),\:\:\left(−\mathrm{7},\:−\mathrm{6}\right),\:\:\left(\mathrm{7},\:−\mathrm{6}\right),\:\:\left(−\mathrm{7},\:\mathrm{6}\right) \\ $$$$\mathrm{2}{a}\:+\:{b}\:=\:\mathrm{20},\:\:−\mathrm{20},\:\:\:\:\:\mathrm{8},\:\:−\mathrm{8}\:\:\mathrm{Answers} \\ $$$$ \\ $$$$\bullet\bullet\blacktriangleright\:{a},\:{b}\:\in\:\mathbb{R} \\ $$$$\infty\:{solutions} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$
Commented by Shrinava last updated on 11/Sep/22
thank you dear ser
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{ser} \\ $$

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