GATE Electrical 2014 Set 1 | Question: 1

Question

Given a system of equations:

                                   $x+2y+2z=b_1$

                                   $5x+y+3z=b_2$ 

Which of the following is true regarding its solutions

• A. The system has a unique solution for any given $b_1$ and $b_2$
• B. The system will have infinitely many solutions for any given $b_1$ and $b_2$
• C. Whether or not a solution exists depends on the given $b_1$ and $b_2$
• D. The system would have no solution for any values of $b_1$ and $b_2$

Answer 1

Coefficient Matrix

\[ A = \begin{bmatrix} 1 & 2 & 2 \\ 5 & 1 & 3 \end{bmatrix} \]

 

\[ \text{rank}(A) = 2 \]

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Check the Augmented Matrix

\[ [A \mid b] = \begin{bmatrix} 1 & 2 & 2 & b_1 \\ 5 & 1 & 3 & b_2 \end{bmatrix} \]

Apply \( R_2 \rightarrow R_2 - 5R_1 \):

\[ \begin{bmatrix} 1 & 2 & 2 & b_1 \\ 0 & -9 & -7 & b_2 - 5b_1 \end{bmatrix} \]

For any values of \( b_1 \) and \( b_2 \):

\[ \text{rank}[A \mid b] = 2 = \text{rank}(A) \]

So the system is always consistent (never has no solution).

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Count Free Variables

\[ \text{Free variables} = n - \text{rank}(A) = 3 - 2 = 1 \]

Answer: Option 2 

\[ \boxed{\text{The system will have infinitely many solutions for any given } b_1 \text{ and } b_2} \]

Reason: 2 equations, 3 unknowns, \( \text{rank}(A) = 2 \) → always 1 free variable → always infinitely many solutions regardless of \( b_1 \) and \( b_2 \).