Question

If the system of linear equations: \[ x + y + 2z = 6, \] \[ 2x + 3y + az = a + 1, \] \[ -x - 3y + bz = 2b, \] where \( a, b \in \mathbb{R} \), has infinitely many solutions, then \( 7a + 3b \) is equal to:

• A. 22
• B. 16
• C. 9
• D. 12

Hint:

For systems of linear equations with infinitely many solutions: - The determinant of the coefficient matrix must be zero, which indicates linear dependence of the equations. - Solve for the values of the parameters \( a \) and \( b \) by setting the determinant equal to zero, and then use these values to find the required quantity.

Answer 1

The Correct Option is C

For the system to have infinitely many solutions, the determinant of the coefficient matrix must be zero, as this will indicate linear dependence. The coefficient matrix is: 

\[\begin{pmatrix} 1 & 1 & 2 \\ 2 & 3 & a \\ -1 & -3 & b \end{pmatrix}.\]

 We compute the determinant of the matrix and solve the equation for the values of \( a \) and \( b \) that make the determinant equal to zero. This yields the values for \( a \) and \( b \). 

Finally, using these values, we calculate \( 7a + 3b = 9 \).