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:
Show Hint
- 22
- 16
- 9
- 12
The Correct Option is C
Solution and Explanation
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 \( 7a + 3b = 9 \).
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