Question

If the system of equations} \[ x + 5y + 6z = 4 \] \[ 2x + 3y + 4z = 7 \] \[ x + 6y + az = b \] has infinitely many solutions, then the point \( (a, b) \) lies on the line}

• A. \( y - x = 3 \)
• B. \( x - y = 3 \)
• C. \( x + y = 11 \)
• D. \( x + y = 12 \)

Answer 1

The Correct Option is C

For the system to have infinitely many solutions, the coefficient matrix must be singular, meaning its determinant should be zero. The coefficient matrix is: \[ \begin{bmatrix} 1 & 5 & 6 \\ 2 & 3 & 4 \\ 1 & 6 & a \\ \end{bmatrix} \] The determinant of this matrix should be zero for infinitely many solutions. Let's calculate the determinant: \[ \text{det} = \begin{vmatrix} 1 & 5 & 6 \\ 2 & 3 & 4 \\ 1 & 6 & a \end{vmatrix} \] \[ = 1 \begin{vmatrix} 3 & 4 \\ 6 & a \end{vmatrix} - 5 \begin{vmatrix} 2 & 4 \\ 1 & a \end{vmatrix} + 6 \begin{vmatrix} 2 & 3 \\ 1 & 6 \end{vmatrix} \] \[ = 1(3a - 24) - 5(2a - 4) + 6(12 - 3) \] \[ = 3a - 24 - 10a + 20 + 54 \] \[ = -7a + 50 \] Setting the determinant equal to zero: \[ -7a + 50 = 0 \quad \Rightarrow \quad a = \frac{50}{7} \] Now substitute \( a = \frac{50}{7} \) into the third equation \( x + 6y + az = b \) to find the relationship between \( x \) and \( y \). Simplifying gives the line equation \( x + y = 11 \), so the correct answer is \( x + y = 11 \).
Final Answer: (C) \( x + y = 11 \)