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(a / b)[1 - (a / x)] + (b / a)[1 - (b / x)] = 1

Ejecutamos productos.

[(a / b) - (a / b)(a / x)] + [(b / a) - (b / a)(b / x)] = 1

[(a / b) - (a² / bx)] + [(b / a) - (b² / ax)] = 1

(a / b) + (b / a) - (a² / bx) - (b² / ax) = 1

Comun denominador entre : b ; a ; bx ; ax = abx

[a(ax) + b(bx) - a²(a) - b²(b)] / (abx) = 1

[a²x + b²x - a³ - b³] / (abx) = 1

a²x + b²x - a³ - b³ = abx

a²x + b²x - abx = a³ + b³

x(a² - ab + b²) = a³ + b³

X = [a³ + b³] / [a² - ab + b²]

Recordemos que :

[a³ + b³] = (a + b)(a² - ab + b²)

X = [(a + b)(a² - ab + b²)] / [(a² - ab + b²)]

Cancelo (a² - ab + b²)

X = a + b

Rta : X = a + b.