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DESCARGA

CAPITULO XVI

Ecuaciones literales de primer grado con una icognita

Ejercicio 143

Resolver las siguientes ecuaciones:

$\begin{matrix} a (x + 1) & = 1 \\ x + 1 & = \frac{1}{a} \\ x & = \frac{1}{a} - 1 \\ x & = \frac{1 - a}{a} \end{matrix}$
$\begin{matrix} a x - 4 & = b x - 2 \\ a x - b x & = 4 - 2 \\ x (a - b) & = 2 \\ x & = \frac{2}{a - b} \end{matrix}$
$\begin{matrix} a x + b^{2} & = a^{2} - b x \\ a x - b x & = a^{2} - b^{2} \\ x (a - b) & = (a + b) (a - b) \\ x & = a + b \end{matrix}$
$\begin{matrix} 3 (2 a - x) + a x & = a^{2} + 9 \\ 6 a - 3 x + a x & = a^{2} + 9 \\ x (a - 3) & = a^{2} - 6 a + 9 \\ x & = \frac{{(a - 3)}^{2}}{(a - 3)} \\ x & = a - 3 \end{matrix}$
$\begin{matrix} a (x + b) + x (b - a) & = 2 b (2 a - x) \\ a x + a b + b x - a x & = 4 a b - 2 b x \\ b x + 2 b x & = 4 a b - a b \\ 3 b x & = 3 a b \\ x & = a \end{matrix}$
$\begin{matrix} {(x - a)}^{2} - {(x + a)}^{2} & = a (a - 7 x) \\ [(x - a) + (x + a)] [(x - a) - (x + a)] & = a^{2} - 7 a x \\ (x - a + x + a) (x - a - x - a) & = a^{2} - 7 a x \\ 2 x (- 2 a) + 7 a x & = a^{2} \\ - 4 a x + 7 a x & = a^{2} \\ 3 a x & = a^{2} \\ x & = \frac{a^{2}}{3 a} \\ x & = \frac{a}{3} \end{matrix}$
$\begin{matrix} a x - a (a + b) & = - x - (1 + a b) \\ a x + x & = a (a + b) - 1 - a b \\ x (a + 1) & = a^{2} + a b - 1 - a b \\ x (a + 1) & = a^{2} - 1 \\ x (a + 1) & = (a + 1) (a - 1) \\ x & = a - 1 \end{matrix}$
$\begin{matrix} a^{2} (a - x) - b^{2} (x - b) & = b^{2} (x - b) \\ a^{2} (a - x) - b^{2} (x - b) - b^{2} (x - b) & = 0 \\ a^{2} (a - x) - 2 b^{2} (x - b) & = 0 \\ a^{3} - a^{2} x - 2 b^{2} x + 2 b^{3} & = 0 \\ a^{3} + 2 b^{3} & = a^{2} x + 2 b^{2} x \\ a^{3} + 2 b^{3} & = x (a^{2} + 2 b^{2}) \\ x & = \frac{a^{3} + 2 b^{3}}{a^{2} + 2 b^{2}} \end{matrix}$
$\begin{matrix} (x + a) (x - b) - (x + b) (x - 2 a) & = b (a - 2) + 3 a \\ x^{2} - b x + a x - a b - (x^{2} - 2 a x + b x - 2 a b) & = a b - 2 b + 3 a \\ x^{2} - b x + a x - a b - x^{2} + 2 a x - b x + 2 a b & = a b - 2 b + 3 a \\ 3 a x - 2 b x + a b & = a b - 2 b + 3 a \\ x (3 a - 2 b) & = 3 a - 2 b \\ x & = 1 \end{matrix}$
$\begin{matrix} x^{2} + a^{2} & = {(a + x)}^{2} - a (a - 1) \\ x^{2} + a^{2} & = a^{2} + 2 a x + x^{2} - a^{2} + a \\ a^{2} - a & = 2 a x \\ a (a - 1) & = 2 a x \\ \frac{a (a - 1)}{2 a} & = x \\ x & = \frac{a - 1}{2} \end{matrix}$
$\begin{matrix} m (n - x) - m (n - 1) & = m (m x - a) \\ m [(n - x) - (n - 1)] & = m (m x - a) \\ n - x - n + 1 & = m x - a \\ a + 1 & = m x + x \\ a + 1 & = x (m + 1) \\ x & = \frac{a + 1}{m + 1} \end{matrix}$
$\begin{matrix} x - a + 2 & = 2 a x - 3 (a + x) - 2 (a - 5) \\ x - a + 2 & = 2 a x - 3 a - 3 x - 2 a + 10 \\ x - 2 a x + 3 x & = - 5 a + a + 10 - 2 \\ 4 x - 2 a x & = 8 - 4 a \\ 2 x (2 - a) & = (2 - a) \\ x & = 2 \end{matrix}$
$\begin{matrix} a (x - a) - 2 b x & = b (b - 2 a - x) \\ a x - a^{2} - 2 b x & = b^{2} - 2 a b - b x \\ a x - 2 b x + b x & = a^{2} - 2 a b + b^{2} \\ a x - b x & = {(a - b)}^{2} \\ x (a - b) & = {(a - b)}^{2} \\ x & = \frac{{(a - b)}^{2}}{a - b} \\ x & = a - b \end{matrix}$
$\begin{matrix} a x + b x & = {(x + a - b)}^{2} - (x - 2 b) (x + 2 a) \\ a x + b x & = x^{2} + 2 a x + a^{2} - 2 a b - 2 b x + b^{2} - (x^{2} + 2 a x - 2 b x - 4 a b) \\ a x + b x & = x^{2} + 2 a x + a^{2} - 2 a b - 2 b x + b^{2} - x^{2} - 2 a x + 2 b x + 4 a b \\ x (a + b) & = a^{2} + 2 a b + b^{2} \\ x & = \frac{{(a + b)}^{2}}{a + b} \\ x & = a + b \end{matrix}$
$\begin{matrix} x (a + b) - 3 - a (a - 2) & = 2 (x - 1) - x (a - b) \\ x (a + b) - 2 (x - 1) + x (a - b) & = 3 + a (a - 2) \\ a x + b x - 2 x + 2 + a x - b x & = 3 + a^{2} - 2 a \\ 2 a x - 2 x & = a^{2} - 2 a + 3 - 2 \\ 2 x (a - 1) & = {(a - 1)}^{2} \\ x & = \frac{{(a - 1)}^{2}}{2 (a - 1)} \\ x & = \frac{a - 1}{2} \end{matrix}$
$\begin{matrix} (m + 4 x) (3 m + x) & = {(2 x - m)}^{2} + m (15 x - m) \\ 3 m^{2} + m x + 12 m x + 4 x^{2} & = 4 x^{2} - 4 m x + m^{2} + 15 m x - m^{2} \\ 3 m^{2} + 13 m x & = 11 m x \\ 13 m x - 11 m x & = - 3 m^{2} \\ 2 m x & = - 3 m^{2} \\ x & = - \frac{3 m^{2}}{2 m} \\ x & = - \frac{3 m}{2} \end{matrix}$
$\begin{matrix} a^{2} (a - x) - a^{2} (a + 1) - b^{2} (b - x) - b (1 - b^{2}) + a (1 + a) & = 0 \\ a^{2} (a - x) - b^{2} (b - x) & = a^{2} (a + 1) - a (1 + a) + b (1 - b^{2}) \\ a^{3} - a^{2} x - b^{3} + b^{2} x & = a (a + 1) [a - 1] + b - b^{3} \\ a^{3} - a^{2} x + b^{2} x & = a (a^{2} - 1) + b \\ a^{3} + (b^{2} - a^{2}) x & = a^{3} - a + b \\ x & = \frac{b - a}{b^{2} - a^{2}} \\ x & = \frac{b - a}{(b + a) (b - a)} \\ x & = \frac{1}{a + b} \end{matrix}$
$\begin{matrix} {(a x - b)}^{2} & = (b x - a) (a + x) - x^{2} (b - a^{2}) + a^{2} + b (1 - 2 b) \\ {(a x - b)}^{2} - (b x - a) (a + x) + x^{2} (b - a^{2}) & = a^{2} + b (1 - 2 b) \\ a^{2} x^{2} - 2 a b x + b^{2} - a b x - b x^{2} + a^{2} + a x + b x^{2} - a^{2} x^{2} & = a^{2} + b - 2 b^{2} \\ - 3 a b x + a x & = b - 2 b^{2} - b^{2} \\ a x (1 - 3 b) & = b (1 - 3 b) \\ x & = \frac{b}{a} \end{matrix}$
$\begin{matrix} {(x + b)}^{2} - {(x - a)}^{2} - {(a + b)}^{2} & = 0 \\ {(x + b)}^{2} - {(x - a)}^{2} & = {(a + b)}^{2} \\ [(x + b) + (x - a)] [(x + b) - (x - a)] & = {(a + b)}^{2} \\ (x + b + x - a) (x + b - x + a) & = {(a + b)}^{2} \\ (2 x + b - a) (a + b) & = {(a + b)}^{2} \\ 2 x + b - a & = a + b \\ 2 x & = a + a \\ 2 x & = 2 a \\ x & = a \end{matrix}$
$\begin{matrix} {(x + m)}^{3} - 12 m^{3} & = - {(x - m)}^{3} + 2 x^{3} \\ {(x + m)}^{3} + {(x - m)}^{3} & = 2 x^{3} + 12 m^{3} \\ [(x + m) + (x - m)] [{(x + m)}^{2} - (x + m) (x - m) + {(x - m)}^{2}] & = 2 (x^{3} + 6 m^{3}) \\ (x + m + x - m) [x^{2} + 2 m x + m^{2} - (x^{2} - m^{2}) + x^{2} - 2 m x + m^{2}] & = 2 (x^{3} + 6 m^{3}) \\ 2 x [2 x^{2} + 2 m^{2} - x^{2} + m^{2}] & = 2 (x^{3} + 6 m^{3}) \\ x (x^{2} + 3 m^{2}) & = x^{3} + 6 m^{3} \\ x^{3} + 3 x m^{2} & = x^{3} + 6 m^{3} \\ x & = \frac{m^{3}}{3 m^{2}} \\ x & = 2 m \end{matrix}$