Tìm các cặp số nguyên x,y sao cho  \(\left| {x - 7...

- Hướng dẫn giải

Phương pháp giải:

Ta cần chú ý: \(\left| a \right| \ge 0\,\,\forall a.\)

Giải chi tiết:

\(\left| {x - 7} \right| + \left| y \right| = 2.\)

Ta có:  \(\left\{ \matrix{\left| {x + 7} \right| \ge 0 \hfill \cr  \left| y \right| \ge 0 \hfill \cr}  \right.\) mà \(x;\,\,y \in Z \Rightarrow \left\{ \matrix{\left| {x - 7} \right| \in {Z^ + } \hfill \cr  \left| y \right| \in {Z^ + } \hfill \cr}  \right.\)

\(\Rightarrow \left| {x - 7} \right| + \left| y \right| = 2\)

\( \Leftrightarrow \left[ \matrix{ \left\{ \matrix{\left| {x - 7} \right| = 0 \hfill \cr  \left| y \right| = 2 \hfill \cr}  \right. \hfill \cr \left\{ \matrix{ \left| {x - 7} \right| = 1 \hfill \cr  \left| y \right| = 1 \hfill \cr}  \right. \hfill \cr  \left\{ \matrix{\left| {x - 7} \right| = 2 \hfill \cr \left| y \right| = 0 \hfill \cr}  \right. \hfill \cr}  \right. \Leftrightarrow \left[ \matrix{\left\{ \matrix{x - 7 = 0 \hfill \cr \left[ \matrix{y = 2 \hfill \cr y =  - 2 \hfill \cr}  \right. \hfill \cr}  \right. \hfill \cr \left\{ \matrix{x - 7 = 1 \hfill \cr  y =  \pm 1 \hfill \cr}  \right. \hfill \cr  \left\{ \matrix{ x - 7 =  - 1 \hfill \cr  y =  \pm 1 \hfill \cr}  \right. \hfill \cr \left\{ \matrix{ \left[ \matrix{  x - 7 = 2 \hfill \cr x - 7 =  - 2 \hfill \cr}  \right. \hfill \cr y = 0 \hfill \cr}  \right. \hfill \cr}  \right. \Leftrightarrow \left[ \matrix{ \left\{ \matrix{x = 7 \hfill \cr  \left[ \matrix{y = 2 \hfill \cr y =  - 2 \hfill \cr}  \right. \hfill \cr}  \right. \hfill \cr  \left\{ \matrix{x = 8 \hfill \cr y =  \pm 1 \hfill \cr}  \right. \hfill \cr \left\{ \matrix{ x = 6 \hfill \cr y =  \pm 1 \hfill \cr}  \right. \hfill \cr  \left\{ \matrix{\left[ \matrix{x = 9 \hfill \cr x = 5 \hfill \cr}  \right. \hfill \cr y = 0 \hfill \cr}  \right. \hfill \cr}  \right.\)

Vậy các cặp số (x; y) nguyên cần tìm là:

\(\left( {x;y} \right) = \left\{ {\left( {7;\,\,2} \right);\left( {7; - 2} \right);\left( {8;\,\,1} \right);\left( {8; - 1} \right);\left( {6;\,\,1} \right);\left( {6; - 1} \right);\left( {9;\,\,0} \right);\left( {5;\,\,0} \right)} \right\}.\)