Cho hệ phương trình: \[\left\{ {\begin{array}{*{20}{c}}{(m - 1)x + y = 2}\\{mx + y = m + 1}\end{array}} \right.\] với m là tham số. Chứng minh rằng với mọi giá trị của m thì hệ phương trình luôn có nghiệm duy nhất (x; y).

\[\begin{array}{l}\left\{ {\begin{array}{*{20}{c}}{(m - 1)x + y = 2}\\{mx + y = m + 1}\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{y = 2 - (m - 1)x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{mx + 2 - (m - 1)x = m + 1}\end{array}} \right.\\\end{array}\]

\[ \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{y = 2 - (m - 1)x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{mx + 2 - mx + x = m + 1}\end{array}} \right.\]

\[ \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{y = 2 - (m - 1)x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{x = m - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}} \right.\]

\[ \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{y = 2 - {{(m - 1)}^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{x = m - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}} \right.\]

Vậy hệ phương trình luôn có nghiệm duy nhất (x; y) = (m – 1; 2 – (m – 1)²).