GATE CS 2004 Q71 — Linear Algebra: Matrices, Determinants, System of Linear Equations

We are given the system of linear equations:

1. $-x + 5y = -1$
2. $x - y = 2$
3. $x + 3y = 3$

Let's solve the first two equations. Adding equation (1) and equation (2):

$$(-x + 5y) + (x - y) = -1 + 2 \implies 4y = 1 \implies y = \frac{1}{4}$$

Substituting $y = \frac{1}{4}$ into equation (2):

$$x - \frac{1}{4} = 2 \implies x = \frac{9}{4}$$

Now, we verify if this solution $(x, y) = \left(\frac{9}{4}, \frac{1}{4}\right)$ satisfies the third equation:

$$\text{LHS} = x + 3y = \frac{9}{4} + 3\left(\frac{1}{4}\right) = \frac{12}{4} = 3 = \text{RHS}$$

Since the unique solution to the first two equations also satisfies the third equation, the system has a unique solution.