espressione,[19/3:(5/4+3/8-5/6)-1/5]{[(1/2+2/15*6/5)-2/5]*10/3}

[19/3 : (5/4+3/8-5/6)-1/5] : x = x : {[(1/2+2/15·6/5)-2/5]·10/3}

==============================================================

Proporzione:

$\small \left[\dfrac{19}{3} : \left(\dfrac{5}{4}+\dfrac{3}{8}-\dfrac{5}{6}\right)-\dfrac{1}{5}\right] : x = x : \left\{\left[\left(\dfrac{1}{2}+\dfrac{2}{\cancel{15}_5}·\dfrac{\cancel6^2}{5}\right)-\dfrac{2}{5}\right]·\dfrac{10}{3}\right\} $

$\small \left[\dfrac{19}{3} : \left(\dfrac{30+9-20}{24}\right)-\dfrac{1}{5}\right] : x = x : \left\{\left[\left(\dfrac{1}{2}+\dfrac{2}{5}·\dfrac{2}{5}\right)-\dfrac{2}{5}\right]·\dfrac{10}{3}\right\} $

$\small \left[\dfrac{19}{3} : \dfrac{19}{24}-\dfrac{1}{5}\right] : x = x : \left\{\left[\left(\dfrac{1}{2}+\dfrac{4}{25}\right)-\dfrac{2}{5}\right]·\dfrac{10}{3}\right\} $

$\small \left[\dfrac{\cancel{19}^1}{\cancel3_1} · \dfrac{\cancel{24}^8}{\cancel{19}_1}-\dfrac{1}{5}\right] : x = x : \left\{\left[\dfrac{1}{2}+\dfrac{4}{25}-\dfrac{2}{5}\right]·\dfrac{10}{3}\right\} $

$\small \left[\dfrac{1}{1} · \dfrac{8}{1}-\dfrac{1}{5}\right] : x = x : \left\{\left[\dfrac{25+8-20}{50}\right]·\dfrac{10}{3}\right\} $

$\small \left[8-\dfrac{1}{5}\right] : x = x : \left\{\dfrac{13}{\cancel{50}_5}·\dfrac{\cancel{10}^1}{3}\right\} $

$\small \left[\dfrac{40-1}{5}\right] : x = x : \left\{\dfrac{13}{5}·\dfrac{1}{3}\right\} $

$\small \dfrac{39}{5} : x = x : \dfrac{13}{15} $

moltiplica i medi tra loro e gli estremi tra loro:

$\small x·x = \dfrac{13}{\cancel{15}_5}·\dfrac{\cancel{39}^{13}}{5}$

$\small x^2 = \dfrac{13}{5}·\dfrac{13}{5}$

$\small x^2 = \dfrac{169}{25}$

radice quadrata di ambo le parti:

$\small \sqrt{x^2} = \sqrt{\dfrac{169}{25}}$

$\small x= \pm\dfrac{13}{5}$