Question

Consider the system of equations

$x+2y+3z=6,$

$4x+5y+6z=\lambda ,$

$7x+8y+9z=24.$

Then, the value of $\lambda$ for which the system has infinite solutions is

• A. $15$
• B. $20$
• C. $16$
• D. $30$

Answer 1

Answer: (A)

Note that $D=\left|\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right|=0\left(R_3\to R_3-2R_2+R_1\right)$
$D_1=\left|\begin{array}{ccc}6& 2& 3\\ \lambda & 5& 6\\ 24& 8& 9\end{array}\right|=-3\left(30-2\lambda \right)$
$D_2=\left|\begin{array}{ccc}1& 6& 3\\ 4& \lambda & 6\\ 7& 24& 9\end{array}\right|=6\left(30-2\lambda \right)$
$D_3=\left|\begin{array}{ccc}1& 2& 6\\ 4& 5& \lambda \\ 7& 8& 24\end{array}\right|=-3\left(30-2\lambda \right)$
Hence, for $\lambda =15$ the system has infinite solutions