x/(ab)+x/(bc)+x/(ac)-1=abc-x(a+b+c)

Lösung

\frac{x}{ab} + \frac{x}{b c} + \frac{x}{a c} - 1 = ab c - x (a + b + c)

x = \frac{a ^{2} b ^{2} c ^{2} + ab c}{c + a + b + a ^{2} b c + a b ^{2} c + ab c ^{2}}; ab c \neq = 0, a \neq = \frac{- 1 - b ^{2} c - b c ^{2} + b ^{4} c ^{2} + 2 b ^{3} c ^{3} + b ^{2} c ^{4} - 2 b ^{2} c - 2 b c ^{2} + 1}{2 b c}, a \neq = \frac{- 1 - b ^{2} c - b c ^{2} - b ^{4} c ^{2} + 2 b ^{3} c ^{3} + b ^{2} c ^{4} - 2 b ^{2} c - 2 b c ^{2} + 1}{2 b c}

Schritte zur Lösung

\frac{x}{ab} + \frac{x}{b c} + \frac{x}{a c} - 1 = ab c - x (a + b + c)

Multipliziere mit dem kleinsten gemeinsamen Multiplikator

c x + a x + b x - ab c = a^{2} b^{2} c^{2} - x (a + b + c) ab c; ab c \neq = 0

Schreibe

a^{2} b^{2} c^{2} - x (a + b + c) ab c

um:

a^{2} b^{2} c^{2} - a^{2} b c x - a b^{2} c x - ab c^{2} x

c x + a x + b x - ab c = a^{2} b^{2} c^{2} - a^{2} b c x - a b^{2} c x - ab c^{2} x; ab c \neq = 0

Verschiebe

ab c

auf die rechte Seite

c x + a x + b x = a^{2} b^{2} c^{2} - a^{2} b c x - a b^{2} c x - ab c^{2} x + ab c; ab c \neq = 0

Subtrahiere

- a^{2} b c x - a b^{2} c x - ab c^{2} x

von beiden Seiten

c x + a x + b x - (- a^{2} b c x - a b^{2} c x - ab c^{2} x) = a^{2} b^{2} c^{2} - a^{2} b c x - a b^{2} c x - ab c^{2} x + ab c - (- a^{2} b c x - a b^{2} c x - ab c^{2} x); ab c \neq = 0

Vereinfache

c x + a x + b x + a^{2} b c x + a b^{2} c x + ab c^{2} x = a^{2} b^{2} c^{2} + ab c; ab c \neq = 0

Faktorisiere

c x + a x + b x + a^{2} b c x + a b^{2} c x + ab c^{2} x : x (c + a + b + a^{2} b c + a b^{2} c + ab c^{2})

x (c + a + b + a^{2} b c + a b^{2} c + ab c^{2}) = a^{2} b^{2} c^{2} + ab c; ab c \neq = 0

Teile beide Seiten durch

c + a + b + a^{2} b c + a b^{2} c + ab c^{2}; ab c \neq = 0, a \neq = \frac{- 1 - b ^{2} c - b c ^{2} + b ^{4} c ^{2} + 2 b ^{3} c ^{3} + b ^{2} c ^{4} - 2 b ^{2} c - 2 b c ^{2} + 1}{2 b c}, a \neq = \frac{- 1 - b ^{2} c - b c ^{2} - b ^{4} c ^{2} + 2 b ^{3} c ^{3} + b ^{2} c ^{4} - 2 b ^{2} c - 2 b c ^{2} + 1}{2 b c}

x = \frac{a ^{2} b ^{2} c ^{2} + ab c}{c + a + b + a ^{2} b c + a b ^{2} c + ab c ^{2}}; ab c \neq = 0, a \neq = \frac{- 1 - b ^{2} c - b c ^{2} + b ^{4} c ^{2} + 2 b ^{3} c ^{3} + b ^{2} c ^{4} - 2 b ^{2} c - 2 b c ^{2} + 1}{2 b c}, a \neq = \frac{- 1 - b ^{2} c - b c ^{2} - b ^{4} c ^{2} + 2 b ^{3} c ^{3} + b ^{2} c ^{4} - 2 b ^{2} c - 2 b c ^{2} + 1}{2 b c}

3 x - 1 = 26

\frac{1}{3} \cdot x + 2 + \frac{3}{5} \cdot x = x

2 (x - 3) + 5 (x - 1) = - 4

\frac{87}{100} q + 11 = - \frac{1}{100} q + 115

\frac{2}{3} (c x + \frac{1}{2}) - \frac{1}{4} = \frac{5}{2}