Comparte esto 👍👍

CAPITULO XXIII

Ecuaciones indeterminadas
Resolución de una ecuación de primer grado con dos incógnitas. Soluciones enteras y positivas

Ejercicio 173

Hallar todas las soluciones enteras y positivas de:

$\begin{matrix} x + y & = 5 \\ y & = 5 - x \\ 5 - x & \Rightarrow P a r a q u e s e a e n t e r o p o s i t i v o 1 \leq y \leq 4 s i e n d o « y « e n t e r o \\ x = 1 & y = 4 \\ x = 2 & y = 3 \\ x = 3 & y = 2 \\ x = 4 & y = 1 \end{matrix}$
$\begin{matrix} 2 x + 3 y & = 37 \\ 2 x & = 37 - 3 y \\ x & = \frac{37 - 3 y}{2} \\ x & = \frac{36 + 1 - 2 y - y}{2} \\ x & = \frac{}{2} + \frac{1}{2} - \frac{2 y}{2} - \frac{y}{2} \\ x & = 18 - y + \frac{1 - y}{2} \\ x - 18 + y & = \frac{1 - y}{2} \\ \frac{1 - y}{2} & \Rightarrow E n t e r o \\ M u l t i p l i c o p o r 3 \\ \frac{3 - 3 y}{2} \\ \frac{3 - 3 y}{2} & = \frac{2}{2} + \frac{1}{2} - \frac{2 y}{2} - \frac{y}{2} \\ = 1 - y + \frac{1 - y}{2} \Rightarrow E n t e r o \\ \frac{1 - y}{2} & = m \\ 1 - y & = 2 m \\ - y & = 2 m - 1 \\ y & = 1 - 2 m \\ R e e m p l a z o e n l a e c u a c i ó n o r i g i n a l \\ 2 x + 3 (1 - 2 m) & = 37 \\ 2 x + 3 - 6 m & = 37 \\ 2 x & = 37 - 3 + 6 m \\ x & = \frac{34 - 6 m}{2} \\ x & = \frac{2 (17 + 3 m)}{2} \\ x & = 17 + 3 m \\ {\begin{matrix} y = 1 - 2 m \\ x = 17 + 3 m \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 0 & y = 1 & x = 17 \\ m = - 1 & y = 3 & x = 14 \\ m = - 2 & y = 5 & x = 11 \\ m = - 3 & y = 7 & x = 8 \\ m = - 4 & y = 9 & x = 5 \\ m = - 5 & y = 11 & x = 2 \end{matrix} \end{matrix}$
$\begin{matrix} 3 x + 5 y & = 43 \\ 3 x & = 43 - 5 y \\ x & = \frac{43 - 5 y}{3} \\ \frac{43 - 5 y}{3} & = \frac{}{3} + \frac{1}{3} - \frac{3 y}{3} - \frac{2 y}{3} \\ x & = 14 - y + \frac{1 - 2 y}{3} \\ x - 14 + y & = \frac{1 - 2 y}{3} \\ \frac{1 - 2 y}{3} & \Rightarrow E n t e r o \\ M u l t i p l i c o p o r 2 \\ \frac{2 - 4 y}{3} & = \frac{2}{3} - \frac{3 y}{3} - \frac{y}{3} \\ = - y + \frac{2 - y}{3} \\ \frac{2 - y}{3} & = m \\ 2 - y & = 3 m \\ - y & = 3 m - 2 \\ y & = 2 - 3 m R e e m p l a z o e n l a e c u a c i o n o r i g i n a l \\ 3 x + 5 (2 - 3 m) & = 43 \\ 3 x + 10 - 15 m & = 43 \\ 3 x & = 43 - 10 + 15 m \\ 3 x & = 33 + 15 m \\ x & = \frac{3 (11 + 5 m)}{3} \\ x & = 11 + 5 m \\ {\begin{matrix} x = 11 + 5 m \\ y = 2 - 3 m \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 0 & x = 11 & y = 2 \\ m = - 1 & x = 6 & y = 5 \\ m = - 2 & x = 1 & y = 8 \end{matrix} \end{matrix}$
$\begin{matrix} x + 3 y & = 9 \\ x & = 9 - 3 y \\ 9 - 3 y & \Rightarrow E n t e r o \\ D i v i d o p a r a 3 \\ 3 - y & \Rightarrow E n t e r o \\ 3 - y & = m \\ - y & = m - 3 \\ y & = 3 - m R e e m p l a z o e n l a e c u a c i ó n o r i g i n a l \\ x + 3 (3 - m) & = 9 \\ x + 9 - 3 m & = 9 \\ x & = 3 m \\ {\begin{matrix} y = 3 - m \\ x = 3 m \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 1 & x = 3 & y = 2 \\ m = 2 & x = 6 & y = 1 \end{matrix} \end{matrix}$
$\begin{matrix} 7 x + 8 y & = 115 \\ 7 x & = 115 - 8 y \\ x & = \frac{115 - 8 y}{7} \\ x & = \frac{}{7} + \frac{3}{7} - \frac{7 y}{7} - \frac{y}{7} \\ x & = 16 - y + \frac{3 - y}{7} \\ \frac{3 - y}{7} & \Rightarrow E n t e r o \\ M u l t i p l i c o p o r 8 \\ \frac{24 - 8 y}{7} & = \frac{24}{7} - \frac{7 y}{7} - \frac{y}{7} \\ = - y + \frac{24 - y}{7} \\ \frac{24 - y}{7} & = m \\ 24 - y & = 7 m \\ - y & = - 24 + 7 m \\ y & = 24 - 7 m R e e m p l a z o e n l a e c u a c i ó n o r i g i n a l \\ 7 x + 8 (24 - 7 m) & = 115 \\ 7 x + 192 - 56 m & = 115 \\ 7 x - 56 m & = - 77 \\ 7 x & = 56 m - 77 \\ x & = \frac{7 (8 m - 11)}{7} \\ x & = 8 m - 11 \\ {\begin{matrix} y = 24 - 7 m \\ x = 8 m - 11 \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 2 & x = 5 & y = 10 \\ m = 3 & x = 13 & y = 3 \end{matrix} \end{matrix}$
$\begin{matrix} 15 x + 7 y & = 136 \\ 7 y & = 136 - 15 x \\ y & = \frac{136 - 15 x}{7} \\ y & = \frac{}{7} + \frac{3}{7} - \frac{x}{7} - \frac{x}{7} \\ y & = 19 - 2 x + \frac{3 - x}{7} \\ \frac{3 - x}{7} & \Rightarrow E n t e r o \\ M u l t i p l i c o p o r 8 \\ \frac{24 - 8 x}{7} & = \frac{24}{7} - \frac{7 x}{7} - \frac{x}{7} \\ = - x + \frac{24 - x}{7} \\ \frac{24 - x}{7} & = m \\ 24 - x & = 7 m \\ - x & = 7 m - 24 \\ x & = 24 - 7 m R e e m p l a z o e n l a e c u a c i ó n i n i c i a l \\ 15 (24 - 7 m) + 7 y & = 136 \\ 360 - 105 m + 7 y & = 136 \\ 7 y & = 136 - 360 + 105 m \\ 7 y & = - 224 + 105 m \\ y & = \frac{7 (15 m - 32)}{7} \\ y & = 15 m - 32 \\ {\begin{matrix} x = 24 - 7 m \\ y = 15 m - 32 \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 3 & x = 3 & y = 13 \end{matrix} \end{matrix}$
$\begin{matrix} x + 5 y & = 24 \\ x & = 24 - 5 y \\ x & = 24 + 1 - 1 - 5 y \\ x & = \frac{}{5} - \frac{1}{5} - \frac{5 y}{5} \\ x + \frac{1}{5} & = 5 - y \\ 5 - y & = m \\ - y & = m - 5 \\ y & = 5 - m R e e m p l a z o e n l a e c u a c i ó n i n i c i a l \\ x + 5 (5 - m) & = 24 \\ x + 25 - 5 m & = 24 \\ x & = 5 m - 1 \\ {\begin{matrix} y = 5 - m \\ x = 5 m - 1 \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 1 & x = 4 & y = 4 \\ m = 2 & x = 9 & y = 3 \\ m = 3 & x = 14 & y = 2 \\ m = 4 & x = 19 & y = 1 \end{matrix} \end{matrix}$
$\begin{matrix} 9 x + 11 y & = 203 \\ 9 x & = 203 - 11 y \\ x & = \frac{203 - 11 y}{9} \\ x & = \frac{}{9} + \frac{5}{9} - \frac{9 y}{9} - \frac{2 y}{9} \\ x & = 22 - y + \frac{5 - 2 y}{9} \\ \frac{5 - 2 y}{9} & \Rightarrow E n t e r o \\ M u l t i p l i c o p o r 5 \\ \frac{25 - 10 y}{9} & = \frac{}{9} + \frac{7}{9} - \frac{9 y}{9} - \frac{y}{9} \\ = 2 - y + \frac{7 - y}{9} \\ \frac{7 - y}{9} & = m \\ 7 - y & = 9 m \\ - y & = 9 m - 7 \\ y & = 7 - 9 m R e e m p l a z o e n l a e c u a c i ó n i n i c i a l \\ 9 x + 11 (7 - 9 m) & = 203 \\ 9 x + 77 - 99 m & = 203 \\ 9 x & = 203 - 77 + 99 m \\ 9 x & = 126 + 99 m \\ x & = \frac{9 (14 + 11 m)}{9} \\ x & = 14 + 11 m \\ {\begin{matrix} y = 7 - 9 m \\ x = 14 + 11 m \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 0 & x = 14 & y = 7 \\ m = - 1 & x = 3 & y = 16 \end{matrix} \end{matrix}$
$\begin{matrix} 5 x + 2 y & = 73 \\ 2 y & = 73 - 5 x \\ y & = \frac{73 - 5 x}{2} \\ y & = \frac{}{2} + \frac{1}{2} - \frac{x}{2} - \frac{x}{2} \\ y & = 36 - 2 x + \frac{1 - x}{2} \\ y - 36 + 2 x & = \frac{1 - x}{2} \\ \frac{1 - x}{2} & = m \\ 1 - x & = 2 m \\ - x & = 2 m - 1 \\ x & = 1 - 2 m R e e m p l a z o e n l a e c u a c i ó n i n i c i a l \\ 5 (1 - 2 m) + 2 y & = 73 \\ 5 - 10 m + 2 y & = 73 \\ 2 y & = 73 - 5 + 10 m \\ 2 y & = 68 + 10 m \\ y & = \frac{2 (34 + 5 m)}{2} \\ y & = 34 + 5 m \\ {\begin{matrix} x = 1 - 2 m \\ y = 34 + 5 m \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 0 & x = 1 & y = 34 \\ m = - 1 & x = 3 & y = 29 \\ m = - 2 & x = 5 & y = 24 \\ m = - 3 & x = 7 & y = 19 \\ m = - 4 & x = 9 & y = 14 \\ m = - 5 & x = 11 & y = 9 \\ m = - 6 & x = 13 & y = 4 \end{matrix} \end{matrix}$
$\begin{matrix} 8 x + 13 y & = 162 \\ 8 x & = 162 - 13 y \\ x & = \frac{162 - 13 y}{8} \\ x & = \frac{}{8} + \frac{2}{} - \frac{8 y}{8} - \frac{5 y}{8} \\ x & = 20 - y + \frac{2 - 5 y}{8} \\ x - 20 + y & = \frac{2 - 5 y}{8} \\ \frac{2 - 5 y}{8} & \Rightarrow E n t e r o \\ M u l t i p l i c o p o r 5 \\ \frac{10 - 25 y}{8} & = \frac{10}{8} - \frac{y}{8} - \frac{y}{8} \\ = - 3 y + \frac{10 - y}{8} \\ \frac{10 - y}{8} & = m \\ 10 - y & = 8 m \\ - y & = 8 m - 10 \\ y & = 10 - 8 m R e e m p l a z a m o s e n l a e c u a c i ó n i n i c i a l \\ 8 x + 13 (10 - 8 m) & = 162 \\ 8 x + 130 - 104 m & = 162 \\ 8 x & = 162 - 130 + 104 m \\ 8 x & = 32 + 104 m \\ x & = \frac{8 (4 + 13 m)}{8} \\ x & = 4 + 13 m \\ {\begin{matrix} y = 10 - 8 m \\ x = 4 + 13 m \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 0 & x = 4 & y = 10 \\ m = 1 & x = 17 & y = 2 \end{matrix} \end{matrix}$
$\begin{matrix} 7 x + 5 y & = 104 \\ 5 y & = 104 - 7 x \\ y & = \frac{104 - 7 x}{5} \\ y & = \frac{}{5} + \frac{4}{5} - \frac{5 x}{5} - \frac{2 x}{5} \\ y & = 20 - x + \frac{4 - 2 x}{5} \\ y - 20 + x & = \frac{4 - 2 x}{5} \\ \frac{4 - 2 x}{5} & \Rightarrow E n t e r o \\ M u l t i p l i c o p o r 3 \\ \frac{12 - 6 x}{5} & = \frac{}{5} + \frac{2}{5} - \frac{5 x}{5} - \frac{x}{5} \\ = 2 - x + \frac{2 - x}{5} \\ \frac{2 - x}{5} & = m \\ 2 - x & = 5 m \\ - x & = 5 m - 2 \\ x & = 2 - 5 m R e e m p l a z o e n l a e c u a c i ó n i n i c i a l \\ 7 (2 - 5 m) + 5 y & = 104 \\ 14 - 35 m + 5 y & = 104 \\ 5 y & = 104 - 14 + 35 m \\ 5 y & = 90 + 35 m \\ y & = \frac{5 (18 + 7 m)}{5} \\ y & = 18 + 7 m \\ {\begin{matrix} x = 12 - 5 m \\ y = 18 + 7 m \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 0 & x = 2 & y = 18 \\ m = - 1 & x = 7 & y = 11 \\ m = - 2 & x = 12 & y = 4 \end{matrix} \end{matrix}$
$\begin{matrix} 10 x + y & = 32 \\ y & = 32 - 10 x \\ y & = 32 + 2 - 2 - 10 x \\ y & = \frac{30}{10} + \frac{2}{} - \frac{10 x}{10} \\ y & = 3 + \frac{1}{5} - x \\ y - \frac{1}{5} & = 3 - x \\ 3 - x & = m \\ - x & = m - 3 \\ x & = 3 - m R e e m p l a z o e n l a e c u a c i ó n o r i g i n a l \\ 10 (3 - m) + y & = 32 \\ 30 - 10 m + y & = 32 \\ y & = 32 - 30 + 10 m \\ y & = 2 + 10 m \\ {\begin{matrix} x = 3 - m \\ y = 2 + 10 m \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 0 & x = 3 & y = 2 \\ m = 1 & x = 2 & y = 12 \\ m = 2 & x = 1 & y = 22 \end{matrix} \end{matrix}$
$\begin{matrix} 9 x + 4 y & = 86 \\ 4 y & = 86 - 9 x \\ y & = \frac{86 - 9 x}{4} \\ y & = \frac{}{4} + \frac{2}{4} - \frac{x}{4} - \frac{x}{4} \\ y & = 21 - 2 x + \frac{2 - x}{4} \\ y - 21 + 2 x & = \frac{2 - x}{4} \\ \frac{2 - x}{4} & = m \\ 2 - x & = 4 m \\ - x & = 4 m - 2 \\ x & = 2 - 4 m R e e m p l a z o e n l a e c u a c i ó n i n i c i a l \\ 9 (2 - 4 m) + 4 y & = 86 \\ 18 - 36 m + 4 y & = 86 \\ 4 y & = 86 - 18 + 36 m \\ 4 y & = 68 + 36 \\ y & = \frac{4 (17 + 9 m)}{4} \\ y & = 17 + 9 m \\ {\begin{matrix} x = 2 - 4 m \\ y = 17 + 9 m \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 0 & x = 2 & y = 17 \\ m = - 1 & x = 6 & y = 8 \end{matrix} \end{matrix}$
$\begin{matrix} 9 x + 11 y & = 207 \\ 9 x & = 207 - 11 y \\ x & = \frac{207 - 11 y}{9} \\ x & = \frac{}{9} + \frac{9}{9} - \frac{9 y}{9} - \frac{2 y}{9} \\ x & = 22 - y + \frac{9 - 2 y}{9} \\ x - 22 + y & = \frac{9 - 2 y}{9} \\ \frac{9 - 2 y}{9} & \Rightarrow E n t e r o \\ M u l t i p l i c o p o r 5 \\ \frac{45 - 10 y}{9} & = \frac{45 - 9 y - y}{9} \\ = \frac{45}{9} - \frac{9 y}{9} - \frac{y}{9} \\ = \frac{45 - y}{9} - y \\ \frac{45 - y}{9} & = m \\ 45 - y & = 9 m \\ - y & = 9 m - 45 \\ y & = 45 - 9 m R e e m p l a z o e n l a e c u a c i ó n i n i c i a l \\ 9 x + 11 (45 - 9 m) & = 207 \\ 9 x + 495 - 99 m & = 207 \\ 9 x & = 207 - 495 + 99 m \\ 9 x & = 99 m - 288 \\ x & = \frac{9 (11 m - 32)}{9} \\ x & = 11 m - 32 \\ {\begin{matrix} y = 45 - 9 m \\ x = 11 m - 32 \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 3 & x = 1 & y = 18 \\ m = 4 & x = 12 & y = 9 \end{matrix} \end{matrix}$
$\begin{matrix} 11 x + 12 y & = 354 \\ 11 x & = 354 - 12 y \\ x & = \frac{354 - 12 y}{11} \\ x & = \frac{}{11} + \frac{2}{11} - \frac{11 y}{11} - \frac{y}{11} \\ x & = 32 - y + \frac{2 - y}{11} \\ x - 32 + y & = \frac{2 - y}{11} \\ \frac{2 - y}{11} & = m \\ 2 - y & = 11 m \\ - y & = 11 m - 2 \\ y & = 2 - 11 m R e e m p l a z a m o s e n l a e c u a c i ó n i n i c i a l \\ 11 x + 12 (2 - 11 m) & = 354 \\ 11 x + 24 - 132 m & = 354 \\ 11 x & = 354 - 24 + 132 m \\ 11 x & = 330 + 132 m \\ x & = \frac{11 (30 + 12 m)}{11} \\ x & = 30 + 12 m \\ {\begin{matrix} y = 2 - 11 m \\ x = 30 + 12 m \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 0 & x = 30 & y = 2 \\ m = - 1 & x = 18 & y = 13 \\ m = - 2 & x = 6 & y = 24 \end{matrix} \end{matrix}$
$\begin{matrix} 10 x + 13 y & = 294 \\ 10 x & = 294 - 13 y \\ x & = \frac{294 - 13 y}{10} \\ x & = \frac{290}{10} + \frac{4}{10} - \frac{10 y}{10} - \frac{3 y}{10} \\ x - 29 + y & = \frac{4 - 3 y}{10} \\ \frac{4 - 3 y}{10} & \Rightarrow E n t e r o \\ M u l t i p l i c o p o r 7 \\ \frac{28 - 21 y}{10} & = \frac{28}{10} - \frac{20 y}{10} - \frac{y}{10} \\ = - 2 y + \frac{28 - y}{10} \\ \frac{28 - y}{10} & = m \\ 28 - y & = 10 m \\ - y & = 10 m - 28 \\ y & = 28 - 10 m R e e m p l a z o e n l a e c u a c i ó n i n i c i a l \\ 10 x + 13 (28 - 10 m) & = 294 \\ 10 x + 364 - 130 m & = 294 \\ 10 x & = 294 - 364 + 130 m \\ 10 x & = 130 m - 70 \\ x & = \frac{10 (13 m - 7)}{10} \\ x & = 13 m - 7 \\ {\begin{matrix} y = 28 - 10 m \\ x = 13 m - 7 \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 1 & x = 6 & y = 18 \\ m = 2 & x = 19 & y = 8 \end{matrix} \end{matrix}$
$\begin{matrix} 11 x + 8 y & = 300 \\ 8 y & = 300 - 11 x \\ y & = \frac{300 - 11 x}{8} \\ y & = \frac{}{8} + \frac{4}{8} - \frac{3 x}{8} - \frac{8 x}{8} \\ y & = 37 - x + \frac{4 - 3 x}{8} \\ y - 37 + x & = \frac{4 - 3 x}{8} \\ \frac{4 - 3 x}{8} & \Rightarrow E n t e r o \\ M u l t i p l i c o p o r 3 \\ \frac{12 - 9 x}{8} & = \frac{12}{8} - \frac{8 x}{8} - \frac{x}{8} \\ = \frac{12 - x}{8} - x \\ \frac{12 - x}{8} & = m \\ 12 - x & = 8 m \\ - x & = 8 m - 12 \\ x & = 12 - 8 m R e e m p l a z o e n l a e c u a c i ó n i n i c i a l \\ 11 (12 - 8 m) + 8 y & = 300 \\ 132 - 88 m + 8 y & = 300 \\ 8 y & = 300 - 132 + 88 m \\ 8 y & = 168 + 88 m \\ y & = \frac{8 (21 + 11 m)}{8} \\ y & = 21 + 11 m \\ {\begin{matrix} x = 12 - 8 m \\ y = 21 + 11 m \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 0 & x = 12 & y = 21 \\ m = 1 & x = 4 & y = 32 \\ m = - 1 & x = 20 & y = 10 \end{matrix} \end{matrix}$
$\begin{matrix} 21 x + 25 y & = 705 \\ 21 x & = 705 - 25 y \\ x & = \frac{705 - 25 y}{21} \\ x & = \frac{}{21} + \frac{12}{21} - \frac{21 y}{21} - \frac{4 y}{21} \\ x & = 33 - y + \frac{12 - 4 y}{21} \\ \frac{12 - 4 y}{21} & \Rightarrow E n t e r o \\ M u l t i p l i c o p o r 16 \\ \frac{192 - 64 y}{21} & = \frac{192}{21} - \frac{}{21} - \frac{y}{21} \\ = - 3 y + \frac{192 - y}{21} \\ \frac{192 - y}{21} & = m \\ 192 - y & = 21 m \\ - y & = 21 m - 192 \\ y & = 192 - 21 m R e e m p l a z o e n l a e c u a c i ó n \\ 21 x + 25 (192 - 21 m) & = 705 \\ 21 x + 4800 - 525 m & = 705 \\ 21 x & = 705 - 4800 + 525 m \\ 21 x & = 4095 + 525 m \\ x & = \frac{21 (25 m - 195)}{21} \\ x & = 25 m - 195 \\ {\begin{matrix} y = 192 - 21 m \\ x = 25 m - 195 \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 8 & x = 5 & y = 24 \\ m = 9 & x = 30 & y = 3 \end{matrix} \end{matrix}$
Hallar la solución general y los tres menores pares de valores enteros positivos x e y que satisfacen las ecuaciones siguientes:
$\begin{matrix} 3 x - 4 y & = 5 \\ 3 x & = 5 + 4 y \\ x & = \frac{5 + 4 y}{3} \\ x & = \frac{2 + 3 + 3 y + y}{3} \\ x & = \frac{2}{3} + \frac{3}{3} + \frac{3 y}{3} + \frac{y}{3} \\ x & = 1 + y + \frac{2 + y}{3} \\ \frac{2 + y}{3} & = m \\ 2 + y & = 3 m \\ y & = 3 m - 2 R e e m p l a z o e n l a e c u a c i ó n i n i c i a l \\ 3 x - 4 (3 m - 2) & = 5 \\ 3 x - 12 m + 8 & = 5 \\ 3 x & = 5 - 8 + 12 m \\ 3 x & = 12 m - 3 \\ x & = \frac{3 (4 m - 1)}{3} \\ x & = 4 m - 1 \\ {\begin{matrix} y = 3 m - 2 \\ x = 4 m - 1 \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 1 & x = 3 & y = 1 \\ m = 2 & x = 7 & y = 4 \\ m = 3 & x = 11 & y = 7 \end{matrix} \end{matrix}$
$\begin{matrix} 5 x - 8 y & = 1 \\ 5 x & = 1 + 8 y \\ x & = \frac{1 + 8 y}{5} \\ \frac{1 + 8 y}{5} & \Rightarrow E n t e r o \\ M u l t i p l i c o p o r 2 \\ \frac{2 + 16 y}{5} & = \frac{2}{5} + \frac{y}{5} + \frac{y}{5} \\ = 3 y + \frac{2 + y}{5} \\ \frac{2 + y}{5} & = m \\ 2 + y & = 5 m \\ y & = 5 m - 2 R e e m p l a z o e n l a e c u a c i ó n i n i c i a l \\ 5 x - 8 (5 m - 2) & = 1 \\ 5 x - 40 m + 16 & = 1 \\ 5 x & = 1 - 16 + 40 m \\ 5 x & = 40 m - 15 \\ x & = \frac{5 (8 m - 3)}{5} \\ x & = 8 m - 3 \\ {\begin{matrix} y = 5 m - 2 \\ x = 8 m - 3 \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 1 & x = 5 & y = 3 \\ m = 2 & x = 13 & y = 8 \\ m = 3 & x = 21 & y = 13 \end{matrix} \end{matrix}$
$\begin{matrix} 7 x - 13 y & = 43 \\ 7 x & = 43 + 13 y \\ x & = \frac{43 + 13 y}{7} \\ x & = \frac{}{7} + \frac{1}{7} + \frac{7 y}{7} + \frac{6 y}{7} \\ x & = 6 + y + \frac{1 + 6 y}{7} \\ x - 6 - y & = \frac{1 + 6 y}{7} \\ \frac{1 + 6 y}{7} & \Rightarrow E n t e r o \\ M u l t i p l i c o p o r 6 \\ \frac{6 + 36 y}{7} & = \frac{6 + 35 y + y}{7} \\ = \frac{6}{7} + \frac{y}{7} + \frac{y}{7} \\ = 5 y + \frac{6 + y}{7} \\ \frac{6 + y}{7} & = m \\ 6 + y & = 7 m \\ y & = 7 m - 6 R e e m p l a z o e n l a e c u a c i ó n o r i g i n a l \\ 7 x - 13 (7 m - 6) & = 43 \\ 7 x - 91 m + 78 & = 43 \\ 7 x & = 43 - 78 + 91 m \\ 7 x & = 91 m - 35 \\ x & = \frac{7 (13 m - 5)}{7} \\ x & = 13 m - 5 \\ {\begin{matrix} y = 7 m - 6 \\ x = 13 m - 5 \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 1 & x = 8 & y = 1 \\ m = 2 & x = 21 & y = 8 \\ m = 3 & x = 34 & y = 15 \end{matrix} \end{matrix}$
$\begin{matrix} 11 x - 12 y & = 0 \\ 11 x & = 12 y \\ x & = \frac{12 y}{11} \\ x & = \frac{11 y + y}{11} \\ x & = \frac{11 y}{11} + \frac{y}{11} \\ x - y & = \frac{y}{11} \\ \frac{y}{11} & = m \\ y & = 11 m R e e m p l a z o e n l a e c u a c i ó n i n i c i a l \\ 11 x - 12 (11 m) & = 0 \\ 11 x - 132 m & = 0 \\ 11 x & = 132 m \\ x & = \frac{m}{11} \\ x & = 12 m \\ {\begin{matrix} x = 12 m \\ y = 11 m \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 1 & x = 12 & y = 11 \\ m = 2 & x = 24 & y = 22 \\ m = 3 & x = 36 & y = 33 \end{matrix} \end{matrix}$
$\begin{matrix} 14 x - 17 y & = 32 \\ 14 x & = 32 + 17 y \\ x & = \frac{32 + 17 y}{14} \\ x & = \frac{28 + 4 + 14 y + 3 y}{14} \\ x & = \frac{}{14} + \frac{4}{14} + \frac{14 y}{14} + \frac{3 y}{14} \\ x & = 2 + y + \frac{4 + 3 y}{14} \\ \frac{4 + 3 y}{14} & \Rightarrow E n t e r o \\ M u l t i p l i c o p o r 5 \\ \frac{20 + 15 y}{14} & = \frac{14 + 6 + 14 y + y}{14} \\ = \frac{14}{14} + \frac{6}{14} + \frac{14 y}{14} + \frac{y}{14} \\ = 1 + y + \frac{6 + y}{14} \\ \frac{6 + y}{14} & = m \\ 6 + y & = 14 m \\ y & = 14 m - 6 R e e m p l a z o e n l a e c u a c i ó n i n i c i a l \\ 14 x - 17 (14 m - 6) & = 32 \\ 14 x - 238 m + 102 & = 32 \\ 14 x & = 32 - 102 + 238 m \\ 14 x & = 238 m - 70 \\ x & = \frac{14 (17 m - 5)}{14} \\ x & = 17 m - 5 \\ {\begin{matrix} y = 14 m - 6 \\ x = 17 m - 5 \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 1 & x = 12 & y = 8 \\ m = 2 & x = 29 & y = 22 \\ m = 3 & x = 46 & y = 36 \end{matrix} \end{matrix}$
$\begin{matrix} 7 x - 11 y & = 83 \\ 7 x & = 83 + 11 y \\ x & = \frac{83 + 11 y}{7} \\ x & = \frac{77 + 6 + 7 y + 4 y}{7} \\ x & = \frac{}{7} + \frac{6}{7} + \frac{7 y}{7} + \frac{4 y}{7} \\ x & = 11 + y + \frac{6 + 4 y}{7} \\ x - 11 - y & = \frac{6 + 4 y}{7} \\ \frac{6 + 4 y}{7} & \Rightarrow E n t e r o \\ M u l t i p l i c o p o r 2 \\ \frac{12 + 8 y}{7} & = \frac{7 + 5 + 7 y + y}{7} \\ = \frac{7}{7} + \frac{5}{7} + \frac{7 y}{7} + \frac{y}{7} \\ = 1 + y + \frac{5 + y}{7} \\ \frac{5 + y}{7} & = m \\ 5 + y & = 7 m \\ y & = 7 m - 5 R e e m p l a z o e n l a e c u a c i ó n i n i c i a l \\ 7 x - 11 (7 m - 5) & = 83 \\ 7 x - 77 m + 55 & = 83 \\ 7 x & = 83 - 55 + 77 m \\ 7 x & = 28 + 77 m \\ x & = \frac{7 (4 + 11 m)}{7} \\ x & = 11 m + 4 \\ {\begin{matrix} y = 7 m - 5 \\ x = 11 m + 4 \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 1 & x = 15 & y = 2 \\ m = 2 & x = 26 & y = 9 \\ m = 3 & x = 37 & y = 16 \end{matrix} \end{matrix}$
$\begin{matrix} 8 x - 13 y & = 407 \\ 8 x & = 407 + 13 y \\ x & = \frac{407 + 13 y}{8} \\ x & = \frac{400 + 7 + 8 y + 5 y}{8} \\ x & = \frac{}{8} + \frac{7}{8} + \frac{8 y}{8} + \frac{5 y}{8} \\ x & = 50 + y + \frac{7 + 5 y}{8} \\ x - 50 - y & = \frac{7 + 5 y}{8} \\ \frac{7 + 5 y}{8} & \Rightarrow E n t e r o \\ M u l t i p l i c o p o r 5 \\ \frac{35 + 25 y}{8} & = \frac{32 + 3 + 24 y + y}{8} \\ = \frac{}{8} + \frac{3}{8} + \frac{y}{8} + \frac{y}{8} \\ = 4 + 3 y + \frac{3 + y}{8} \\ \frac{3 + y}{8} & = m \\ 3 + y & = 8 m \\ y & = 8 m - 3 R e e m p l a z o e n l a e c u a c i ó n i n i c i a l \\ 8 x - 13 (8 m - 3) & = 407 \\ 8 x - 104 m + 39 & = 407 \\ 8 x & = 407 - 39 + 104 m \\ 8 x & = 368 + 104 m \\ x & = \frac{8 (13 m + 46)}{8} \\ x & = 13 m + 46 \\ {\begin{matrix} y = 8 m - 3 \\ x = 13 m + 46 \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 1 & x = 59 & y = 5 \\ m = 2 & x = 72 & y = 13 \\ m = 3 & x = 85 & y = 21 \end{matrix} \end{matrix}$
$\begin{matrix} 20 y - 23 x & = 411 \\ 20 y & = 23 x + 411 \\ y & = \frac{23 x + 411}{20} \\ y & = \frac{20 x + 3 x + 400 + 11}{20} \\ y & = \frac{20 x}{20} + \frac{3 x}{20} + \frac{x}{20} + \frac{3 x}{20} \\ y & = 20 + x + \frac{11 + 3 x}{20} \\ y - 20 - x & = \frac{11 + 3 x}{20} \\ \frac{11 + 3 x}{20} & \Rightarrow E n t e r o \\ M u l t i p l i c o p o r 7 \\ \frac{77 + 21 x}{20} & = \frac{60 + 17 + 20 x + x}{20} \\ = \frac{}{20} + \frac{17}{20} + \frac{20 x}{20} + \frac{x}{20} \\ = 3 + x + \frac{17 + x}{20} \\ \frac{17 + x}{20} & = m \\ 17 + x & = 20 m \\ x & = 20 m - 17 R e e m p l a z o e n l a e c u a c i ó n i n i c i a l \\ 20 y - 23 (20 m - 17) & = 411 \\ 20 y - 460 m + 391 & = 411 \\ 20 y & = 411 - 391 + 460 m \\ 20 y & = 20 + 460 m \\ y & = \frac{20 (1 + 23 m)}{20} \\ y & = 23 m + 1 \\ {\begin{matrix} x = 20 m - 17 \\ y = 23 m + 1 \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 1 & x = 3 & y = 24 \\ m = 2 & x = 23 & y = 47 \\ m = 3 & x = 43 & y = 70 \end{matrix} \end{matrix}$
$\begin{matrix} 5 y - 7 x & = 312 \\ 5 y & = 312 + 7 x \\ y & = \frac{312 + 7 x}{5} \\ y & = \frac{310 + 2 + 5 x + 2 x}{5} \\ y & = \frac{}{5} + \frac{2}{5} + \frac{5 x}{5} + \frac{2 x}{5} \\ y & = 62 + x + \frac{2 + 2 x}{5} \\ y - 62 - x & = \frac{2 + 2 x}{5} \\ \frac{2 + 2 x}{5} & \Rightarrow E n t e r o \\ M u l t i p l i c o p o r 3 \\ \frac{6 + 6 x}{5} & = \frac{5 + 1 + 5 x + x}{5} \\ = \frac{5}{5} + \frac{1}{5} + \frac{5 x}{5} + \frac{x}{5} \\ = 1 + x + \frac{1 + x}{5} \\ \frac{1 + x}{5} & = m \\ 1 + x & = 5 m \\ x & = 5 m - 1 R e e m p l a z o e n l a e c u a c i ó n i n i c i a l \\ 5 y - 7 (5 m - 1) & = 312 \\ 5 y - 35 m + 7 & = 312 \\ 5 y & = 312 - 7 + 35 m \\ 5 y & = 305 + 35 m \\ y & = \frac{5 (7 m + 61)}{5} \\ y & = 7 m + 61 \\ {\begin{matrix} x = 5 m - 1 \\ y = 7 m + 61 \end{matrix} \\ P a r a q u e e l s i s t e m a d e e c u a c i o n e s s e a e n t e r o p o s i t i v o m, \\ t i e n e q u e t o m a r l o s s i g u i e n t e s v a l o r e s \\ \begin{matrix} m = 1 & x = 4 & y = 68 \\ m = 2 & x = 9 & y = 75 \\ m = 3 & x = 14 & y = 82 \end{matrix} \end{matrix}$