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DESCARGA

CAPITULO XVI

Ecuaciones literales de primer grado con una icognita

Ejercicio 144

Resolver las siguientes ecuaciones:

$\begin{matrix} \frac{m}{x} - \frac{1}{m} & = \frac{2}{m} \\ \frac{m^{2} - x}{m x} & = \frac{2}{m} \\ m^{2} - x & = 2 x \\ m^{2} & = 2 x + x \\ m^{2} & = 3 x \\ x & = \frac{m^{2}}{3} \end{matrix}$
$\begin{matrix} \frac{a}{x} + \frac{b}{2} & = \frac{4 a}{x} \\ \frac{2 a + b x}{2 x} & = \frac{4 a}{x} \\ 2 a + b x & = 8 a \\ b x & = 8 a - 2 a \\ b x & = 6 a \\ x & = \frac{6 a}{b} \end{matrix}$
$\begin{matrix} \frac{x}{2 a} - \frac{1 - x}{a^{2}} & = \frac{1}{2 a} \\ \frac{a x - 1 + x}{2 a^{2}} & = \frac{1}{2 a} \\ a x - 1 + x & = a \\ x (a + 1) & = a + 1 \\ x & = 1 \end{matrix}$
$\begin{matrix} \frac{m}{x} + \frac{n}{m} & = \frac{n}{x} + 1 \\ \frac{m^{2} + n x}{m x} & = \frac{n + x}{x} \\ m^{2} + n x & = m (n + x) \\ m^{2} + n x & = m n + m x \\ m^{2} - m n & = m x - n x \\ m (m - n) & = x (m - n) \\ x & = m \end{matrix}$
$\begin{matrix} \frac{a - 1}{a} + \frac{1}{2} & = \frac{3 a - 2}{x} \\ \frac{2 (a - 1) + a}{2 a} & = \frac{3 a - 2}{x} \\ \frac{2 a - 2 + a}{2 a} & = \frac{3 a - 2}{x} \\ \frac{3 a - 2}{2 a} & = \frac{3 a - 2}{x} \\ x (3 a - 2) & = 2 a (3 a - 2) \\ x & = 2 a \end{matrix}$
$\begin{matrix} \frac{a - x}{a} - \frac{b - x}{b} & = \frac{2 (a - b)}{a b} \\ \frac{b (a - x) - a (b - x)}{a b} & = \frac{2 (a - b)}{a b} \\ a b - b x - a b + a x & = 2 (a - b) \\ x (a - b) & = 2 (a - b) \\ x & = 2 \end{matrix}$
$\begin{matrix} \frac{x - 3 a}{a^{2}} - \frac{2 a - x}{a b} & = - \frac{1}{a} \\ \frac{b (x - 3 a) - a (2 a - x)}{a^{2} b} & = - \frac{1}{a} \\ b x - 3 a b - 2 a^{2} + a x & = - a b \\ a x + b x & = 2 a^{2} + 3 a b - a b \\ a x + b x & = 2 a^{2} + 2 a b \\ x (a + b) & = 2 a (a + b) \end{matrix}$
$\begin{matrix} \frac{x + m}{m} - \frac{x + n}{n} & = \frac{m^{2} + n^{2}}{m n} - 2 \\ \frac{n (x + m) - m (x + n)}{m n} & = \frac{m^{2} + n^{2} - 2 m n}{m n} \\ n x + m n - m x - m n & = n^{2} - 2 m n + m^{2} \\ x (n - m) & = {(n - m)}^{2} \\ x & = n - m \end{matrix}$
$\begin{matrix} \frac{x - b}{a} & = 2 - \frac{x - a}{b} \\ \frac{x - b}{a} + \frac{x - a}{b} & = 2 \\ \frac{b (x - b) + a (x - a)}{a b} & = 2 \\ b x - b^{2} + a x - a^{2} & = 2 a b \\ a x + b x & = a^{2} + 2 a b + b^{2} \\ x (a + b) & = {(a + b)}^{2} \\ x & = a + b \end{matrix}$
$\begin{matrix} \frac{4 x}{2 a + b} - 3 & = - \frac{3}{2} \\ \frac{4 x}{2 a + b} & = 3 - \frac{3}{2} \\ \frac{4 x}{2 a + b} & = \frac{6 - 3}{2} \\ \frac{4 x}{2 a + b} & = \frac{3}{2} \\ 2 (4 x) & = 3 (2 a + b) \\ 8 x & = 3 (2 a + b) \\ x & = \frac{3 (2 a + b)}{8} \end{matrix}$
$\begin{matrix} \frac{2 a + 3 x}{x + a} & = \frac{2 (6 x - a)}{4 x + a} \\ \frac{2 a + 3 x}{x + a} & = \frac{12 x - 2 a}{4 x + a} \\ (2 a + 3 x) (4 x + a) & = (12 x - 2 a) (x + a) \\ 8 a x + 2 a^{2} + 12 x^{2} + 3 a x & = 12 x^{2} + 12 a x - 2 a x - 2 a^{2} \\ 2 a^{2} + 2 a^{2} & = 10 a x - 11 a x \\ 4 a^{2} & = - a x \\ x & = - 4 a \end{matrix}$
$\begin{matrix} \frac{2 (x - c)}{4 x - b} & = \frac{2 x + c}{4 (x - b)} \\ \frac{2 x - 2 c}{4 x - b} & = \frac{2 x + c}{4 x - 4 b} \\ (2 x - 2 c) (4 x - 4 b) & = (2 x + c) (4 x - b) \\ 8 x^{2} - 8 b x - 8 c x + 8 b c & = 8 x^{2} - 2 b x + 4 c x - b c \\ - 8 b x - 8 c x - 4 c x + 2 b x & = - 8 b c - b c \\ - 6 b x - 12 c x & = - 9 b c \\ - x (b + c) & = - b c \\ x & = \frac{3 b c}{2 (b + c)} \end{matrix}$
$\begin{matrix} \frac{1}{n} - \frac{m}{x} & = \frac{1}{m n} - \frac{1}{x} \\ \frac{x - m n}{n x} & = \frac{x - m n}{m n x} \\ m (x - m n) & = x - m n \\ m x - m^{2} n & = x - m n \\ m x - x & = m^{2} n - m n \\ x (m - 1) & = m n (m - 1) \\ x & = m n \end{matrix}$
$\begin{matrix} \frac{(x - 2 b) (2 x + a)}{(x - a) (a - 2 b + x)} & = 2 \\ (x - 2 b) (2 x + a) & = 2 (x - a) (a - 2 b + x) \\ 2 x^{2} + a x - 4 b x - 2 a b & = 2 (a x - 2 b x + x^{2} - a^{2} + 2 a b - a x) \\ 2 x^{2} + a x - 4 b x - 2 a b & = - 4 b x + 2 x^{2} - 2 a^{2} + 4 a b \\ a x & = 2 a b - 2 a^{2} + 4 a b \\ a x & = 6 a b - 2 a^{2} \\ a x & = 2 a (3 b - a) \\ x & = 2 (3 b - a) \end{matrix}$
$\begin{matrix} \frac{x + m}{x - n} & = \frac{n + x}{m + x} \\ (x + m) (m + x) & = (n + x) (x - n) \\ {(x + m)}^{2} & = x^{2} - n^{2} \\ x^{2} + 2 m x + m^{2} & = x^{2} - n^{2} \\ 2 m x & = - m^{2} - n^{2} \\ x & = - \frac{m^{2} + n^{2}}{2 m} \end{matrix}$
$\begin{matrix} \frac{x (2 x + 3 b) (x + b)}{x + 3 b} & = 2 x^{2} - b x + b^{2} \\ x (2 x^{2} + 3 b x + 2 b x + 3 b^{2}) & = (x + 3 b) (2 x^{2} - b x + b^{2}) \\ 2 x^{3} + 5 b x^{2} + 3 b^{2} x & = 2 x^{3} - b x^{2} + b^{2} x + 6 b x^{2} - 3 b^{2} x + 3 b^{3} \\ 3 b^{2} x + 2 b^{2} x & = 3 b^{3} \\ 5 b^{2} x & = 3 b^{3} \\ x & = \frac{3 b}{5} \end{matrix}$
$\begin{matrix} \frac{3}{4} (\frac{x}{b} + \frac{x}{a}) & = \frac{1}{3} (\frac{x}{b} - \frac{x}{a}) + \frac{5 a + 13 b}{12 a} \\ \frac{3}{4} (\frac{a x + b x}{a b}) & = \frac{1}{3} [\frac{x}{b} - \frac{x}{a} + \frac{5 a + 13 b}{4 a}] \\ \frac{3}{4} (\frac{a x + b x}{a b}) & = \frac{1}{3} [\frac{4 a x - 4 b x + 5 a b + 13 b^{2}}{4 a b}] \\ 9 (a x + b x) & = 4 a x - 4 b x + 5 a b + 13 b^{2} \\ 9 a x + 9 b x - 4 a x + 4 b x & = 5 a b + 13 b^{2} \\ 5 a x + 13 b x & = b (5 a + 13 b) \\ x (5 a + 13 b) & = b (5 a + 13 b) \\ x & = b \end{matrix}$
$\begin{matrix} \frac{x + a}{3} & = \frac{{(x - b)}^{2}}{3 x - a} + \frac{3 a b - 3 b^{2}}{9 x - 3 a} \\ \frac{x + a}{3} & = \frac{{(x - b)}^{2}}{3 x - a} + \frac{3 (a b - b^{2})}{3 (3 x - a)} \\ \frac{x + a}{3} & = \frac{1}{3 x - a} [{(x - b)}^{2} + a b - b^{2}] \\ \frac{x + a}{3} & = \frac{1}{3 x - a} [x^{2} - 2 b x + b^{2} + a b - b^{2}] \\ (x + a) (3 x - a) & = 3 (x^{2} - 2 b x + a b) \\ 3 x^{2} - a x + 3 a x - a^{2} & = 3 x^{2} - 6 b x + 3 a b \\ 2 a x + 6 b x & = a^{2} + 3 a b \\ 2 x (a + 3 b) & = a (a + 3 b) \\ x & = \frac{a}{2} \end{matrix}$
$\begin{matrix} \frac{5 x + a}{3 x + b} & = \frac{5 x - b}{3 x - a} \\ (5 x + a) (3 x - a) & = (5 x - b) (3 x + b) \\ 15 x^{2} - 5 a x + 3 a x - a^{2} & = 15 x^{2} + 5 b x - 3 b x - b^{2} \\ - 5 a x + 3 a x - 5 b x + 3 b x & = a^{2} - b^{2} \\ - 2 a x - 2 b x & = (a - b) (a + b) \\ - 2 x (a + b) & = (a - b) (a + b) \\ x & = \frac{b - a}{2} \end{matrix}$
$\begin{matrix} \frac{x + a}{x - a} - \frac{x - a}{x + a} & = \frac{a (2 x + a b)}{x^{2} - a^{2}} \\ \frac{{(x + a)}^{2} - {(x - a)}^{2}}{(x - a) (x + a)} & = \frac{a (2 x + a b)}{x^{2} - a^{2}} \\ [(x + a) + (x - a)] [(x + a) - (x - a)] & = 2 a x + a^{2} b \\ (x + a + x - a) (x + a - x + a) & = 2 a x + a^{2} b \\ 2 x (2 a) & = 2 a x + a^{2} b \\ 4 a x - 2 a x & = a^{2} b \\ 2 a x & = a^{2} b \\ x & = \frac{a b}{2} \end{matrix}$
$\begin{matrix} \frac{2 x - 3 a}{x + 4 a} - 2 & = \frac{11 a}{x^{2} - 16 a^{2}} \\ \frac{2 x - 3 a - 2 (x + 4 a)}{x + 4 a} & = \frac{11 a}{(x + 4 a) (x - 4 a)} \\ 2 x - 3 a - 2 x - 8 a & = \frac{11 a}{x - 4 a} \\ - 11 a (x - 4 a) & = 11 a \\ - x + 4 a & = 1 \\ x & = 4 a - 1 \end{matrix}$
$\begin{matrix} \frac{1}{x + a} + \frac{x^{2}}{a^{2} + a x} & = \frac{x + a}{a} \\ \frac{1}{x + a} + \frac{x^{2}}{a (x + a)} & = \frac{x + a}{a} \\ \frac{1}{x + a} (1 + \frac{x^{2}}{a}) & = \frac{x + a}{a} \\ \frac{1}{x + a} (\frac{a + x^{2}}{a}) & = \frac{x + a}{a} \\ a + x^{2} & = {(x + a)}^{2} \\ a + x^{2} & = x^{2} + 2 a x + a^{2} \\ a - a^{2} & = 2 a x \\ a (1 - a) & = 2 a x \\ x & = \frac{1 - a}{2} \end{matrix}$
$\begin{matrix} \frac{2 (a + x)}{b} - \frac{3 (b + x)}{a} & = \frac{6 (a^{2} - 2 b^{2})}{a b} \\ \frac{2 a (a + x) - 3 b (b + x)}{a b} & = \frac{6 (a^{2} - 2 b^{2})}{a b} \\ 2 a^{2} + 2 a x - 3 b^{2} - 3 b x & = 6 a^{2} - 12 b^{2} \\ 2 a x - 3 b x & = 6 a^{2} - 2 a^{2} - 12 b^{2} + 3 b^{2} \\ x (2 a - 3 b) & = 4 a^{2} - 9 b^{2} \\ x (2 a - 3 b) & = (2 a + 3 b) (2 a - 3 b) \\ x & = 2 a + 3 b \end{matrix}$
$\begin{matrix} m (n - x) - (m - n) (m + x) & = n^{2} - \frac{1}{n} (2 m n^{2} - 3 m^{2} n) \\ m n - m x - (m^{2} + m x - m n - n x) & = n^{2} - \frac{1}{n} n (2 m n - 3 m^{2}) \\ m n - m x - m^{2} - m x + m n + n x & = n^{2} - 2 m n + 3 m^{2} \\ n x - 2 m x & = n^{2} - 2 m n + 3 m^{2} + m^{2} \\ x (n - 2 m) & = n^{2} - 2 m n + 4 m^{2} \\ x (n - 2 m) & = {(n - 2 m)}^{2} \\ x & = n - 2 m \end{matrix}$