Example 1: Solve the system:
x + y = 5, x - y = 1.

Solution: Add equations → 2x = 6 → x = 3, then y = 2. Unique solution: (3, 2).

Example 2: 2x + y = 10, x - y = 1. Substitution: x = y+1 → 2(y+1)+y=10 → 3y+2=10 → y=8/3, x=11/3. Unique solution.

Example 3: x + y = 2, 2x + 2y = 4. Both are multiples → infinite solutions. Rank(A)=1, Rank([A|B])=1, variables=2 → infinite solutions.

Example 4: x + y = 2, x + y = 3. Contradiction → No solution. Rank(A)=1, Rank([A|B])=2.

Example 5: 2x + 3y = 12, 3x + 4y = 17. Elimination: Multiply first eqn by 3, second by 2 → 6x+9y=36, 6x+8y=34 → subtract → y=2 → x=3. Solution: (3,2).

Example 6: Homogeneous system: x + y = 0, 2x + 2y = 0. Infinite solutions since both are multiples. Trivial solution (0,0) also exists.

Example 7: Homogeneous: x - y = 0, y - z = 0. Solutions: x=y=z. Infinite solutions.

Example 8: Non-homogeneous: x + y = 4, x - y = 0. Solve → from 2nd eqn: x=y. Substitute → 2x=4 → x=2, y=2. Unique solution.

Example 9: 2x + y + z = 4, x - y + z = 2, x + y - z = 1. Solving gives x=1, y=1, z=1. Unique solution.

Example 10: 2x + 3y = 6, 4x + 6y = 12. Both equations dependent → infinite solutions.

Example 11: 2x + 3y = 6, 4x + 6y = 15. Inconsistent → no solution.

Example 12: 3x - y = 7, 2x + y = 8. Add → 5x = 15 → x=3, y=2. Unique solution.

Example 13: 2x - y + z = 3, x + y + z = 6, x - y + 2z = 5. Solving → x=2, y=1, z=3. Unique solution.

Example 14: Homogeneous: x + y + z = 0, 2x + 2y + 2z = 0. Infinite solutions (plane equation).

Example 15: Non-homogeneous: x + y = 2, 2x + 2y = 5. Contradiction → no solution.

Example 11: 3×3 System (Unique Solution)

Solve the system:
x + y + z = 6
2x – y + z = 3
x + 2y – z = 3

Solution:

Coefficient matrix A = $\left[\begin{array}{ccc}1& 1& 1\\ 2& -1& 1\\ 1& 2& -1\end{array}\right]$ , B = (6, 3, 3)^T.

By elimination:
→ z = 2, y = 1, x = 3.
Solution: (x, y, z) = (3, 1, 2).

Example 12: 3×3 System (Infinite Solutions)

Solve the system:
x + y + z = 4
2x + 2y + 2z = 8
3x + 3y + 3z = 12

Solution:

All equations are multiples of the first → rank(A) = 1 < 3. So infinite solutions exist.

General solution: x = t, y = s, z = 4 – t – s, where t, s ∈ ℝ.

Example 13: 3×3 System (No Solution)

Solve the system:
x + y + z = 1
2x + 2y + 2z = 2
x + y + z = 5

Solution:

First and third equations contradict (same LHS, different RHS). → rank(A) ≠ rank([A|B]). No solution (inconsistent system).

Example 14: 4×4 System (Unique Solution)

Solve the system:
x + y + z + w = 10
2x – y + z + w = 8
x + 2y – z + w = 6
x + y + z – w = 4

Solution (sketch):

Coefficient matrix is 4×4 and invertible → rank = 4. Solving step by step: → (x, y, z, w) = (3, 2, 1, 4).

Example 15: 4×4 System (Infinite Solutions)

Solve the system:
x + y + z + w = 5
2x + 2y + 2z + 2w = 10
3x + 3y + 3z + 3w = 15
4x + 4y + 4z + 4w = 20

Solution:

All equations are multiples of the first → rank(A) = 1 < 4. → Infinite solutions exist. General solution: x = t₁, y = t₂, z = t₃, w = 5 – t₁ – t₂ – t₃.