Determine whether the system x + y = 2 and 2x + 2y = 5 is consistent or inconsistent.

Step 1: Write the augmented matrix.

The system becomes:
[[1, 1, 2], [2, 2, 5]]

Step 2: Eliminate the first entry in row 2.

Use:
R2 → R2 - 2R1

Calculate the new row:
2 - 2(1) = 0
2 - 2(1) = 0
5 - 2(2) = 1

The matrix becomes:
[[1, 1, 2], [0, 0, 1]]

Step 3: Interpret the second row.

The second row represents:
0x + 0y = 1

or simply:
0 = 1

This is impossible.

Step 4: State the conclusion.

Because row reduction produces a contradiction, the system has no solution.

A system with no solution is called inconsistent.

Therefore, the system is inconsistent.