Solve the system using row reduction: x + 2y = 7 and 3x - y = 7.

Step 1: Write the augmented matrix.

The system:
x + 2y = 7
3x - y = 7

becomes:
[[1, 2, 7], [3, -1, 7]]

Step 2: Eliminate the x-entry in the second row.

Use the row operation:
R2 → R2 - 3R1

Calculate the new second row:
3 - 3(1) = 0
-1 - 3(2) = -7
7 - 3(7) = -14

The matrix becomes:
[[1, 2, 7], [0, -7, -14]]

Step 3: Make the second pivot equal to 1.

Divide row 2 by -7:
R2 → (-1/7)R2

The matrix becomes:
[[1, 2, 7], [0, 1, 2]]

Step 4: Eliminate the y-entry in row 1.

Use:
R1 → R1 - 2R2

The first row becomes:
[1, 0, 3]

So the reduced matrix is:
[[1, 0, 3], [0, 1, 2]]

Step 5: Read the solution.

x = 3
y = 2

Therefore, the solution is (3, 2).