GATE CS 1998 Q1.2 — Linear Algebra: Matrices, Determinants, System of Linear Equations

To determine the nature of the solutions for the given system of linear equations, let us analyze the equations:

1. $x + 2y = 5$  — (Equation 1)
2. $4x + 8y = 12$  — (Equation 2)
3. $3x + 6y + 3z = 15$  — (Equation 3)

Let us simplify Equation 2 by dividing both sides by $4$:

Now, comparing this simplified equation with Equation 1:

• From Equation 1: $x + 2y = 5$
• From Equation 2: $x + 2y = 3$

Since $x + 2y$ cannot simultaneously equal both $5$ and $3$, these two equations are inconsistent and contradict each other. Therefore, there exists no set of values for $x$, $y$, and $z$ that can satisfy all three equations simultaneously.

Thus, the system has no solution.