Give an example of a system of two linear equations in two unknowns which has no solution.
Solution
1. The system\[\begin{cases} x + y = 1 \\ x + y = 2 \end{cases}\]has no solution.
the two equations are inconsistent.
2. Justification: subtracting the first equation from the second gives \(0 = 1\), a contradiction.
\((x + y) - (x + y) = 2 - 1\), i.e. \(0 = 1\).
3. Hence no pair \((x, y)\) satisfies both equations.
2.
4. Q.E.D.
3.