Question

solve 3^x+1/3y=10/3

answer x=±1​

Answer 1

$\textbf{Given:}$

$3^x+\dfrac{1}{3^x}=\dfrac{10}{3}$

$\textbf{To find:}\;x$

$3^x+\dfrac{1}{3^x}=\dfrac{10}{3}$

$\text{Take }\;t=3^x$

$t+\dfrac{1}{t}=\dfrac{10}{3}$

$\dfrac{t^2+1}{t}=\dfrac{10}{3}$

$3(t^2+1)=10t$

$3t^2+3=10t$

$3t^2-10t+3=0$

$3t^2-9t-t+3=0$

$3t(t-3)-1(t-3)=0$

$(3t-1)(t-3)=0$

$t=3,\frac{1}{3}$

$\text{(i)when t=3}$

$\implies\;3^x=3$

$\implies\;3^x=3^1$

$\implies\bf\;x=1$

$\text{(ii)when t= \frac{1}{3} }$

$\implies\;3^x=\frac{1}{3}$

$\implies\;3^x=3^{-1}$

$\implies\bf\;x=-1$

$\therefore\boxed{\bf\textbf{The solution is}\;x={\pm}1}$

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Answer 2

±1 is the required anwer