The system of linear equations:
2x + 3y + z = 5
4x + 6y + 2z = 10
3x + 5y + z = 8

has:

Question

The system of linear equations:
2x + 3y + z = 5
4x + 6y + 2z = 10
3x + 5y + z = 8

has:

Accepted Answer

Setup: We use the rank method to determine consistency and uniqueness of solutions for a system of linear equations.

Solution:
Write the augmented matrix [A|B]:
⎡2  3  1 | 5⎤
⎢4  6  2 | 10⎥
⎣3  5  1 | 8⎦

Row operations:
R₂ → R₂ - 2R₁:
⎡2  3  1 | 5⎤
⎢0  0  0 | 0⎥
⎣3  5  1 | 8⎦

R₃ → R₃ - (3/2)R₁:
⎡2  3  1 | 5⎤
⎢0  0  0 | 0⎥
⎣0  1/2  -1/2 | 1/2⎦

Swap R₂ and R₃:
⎡2  3  1 | 5⎤
⎢0  1/2  -1/2 | 1/2⎥
⎣0  0  0 | 0⎦

Rank of coefficient matrix A = 2
Rank of augmented matrix [A|B] = 2
n (number of variables) = 3

Since rank(A) = rank([A|B]) < n, the system is consistent but has infinitely many solutions (one free variable).

Trap explanation: Students often confuse the condition for unique solutions (rank = n) with infinitely many solutions (rank < n). Some may incorrectly conclude no solution exists if they miscount the rank.

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