GATE CS 2014 Set 1 Q4 — Linear Algebra: Matrices, Determinants, System of Linear Equations

To find the number of solutions for the given system of linear equations, we can solve a subset of the equations and check if the solution satisfies the remaining equation.

The system of equations is:

1. $3x + 2y = 1$
2. $4x + 7z = 1$
3. $x + y + z = 3$
4. $x - 2y + 7z = 0$

Let's find a candidate solution using equations (1), (2), and (3):

From equation (3), express $z$ in terms of $x$ and $y$:

Substitute $z$ into equation (2):

Now, subtract equation (1) from equation (5):

Substitute $y = 3.8$ back into equation (1):

Now, find $z$ using equation (3):

So, the unique candidate solution is $(x, y, z) = (-2.2, 3.8, 1.4)$.

Let's verify if this candidate solution satisfies the remaining equation (4):

Since the candidate solution satisfies all four equations, the system is consistent and has exactly 1 unique solution.