Question

Solve for $x$ : $\left|\begin{array}{ccc}x-2& 2x-3& 3x-4\\ x-4& 2x-9& 3x-16\\ x-8& 2x-27& 3x-64\end{array}\right|=0$

• A. $\frac{7}{4}$
• B. $\frac{28}{3}$
• C. $\frac{28}{13}$
• D. $\frac{14}{9}$

Answer 1

The correct option is C $\frac{28}{13}$

$\left|\begin{array}{ccc}x-2& 2x-3& 3x-4\\ x-4& 2x-9& 3x-16\\ x-8& 2x-27& 3x-64\end{array}\right|=0$

Using $R_3\to R_3-R_2,R_2\to R_2-R_1$

$\left|\begin{array}{ccc}x-2& 2x-3& 3x-4\\ -2& -6& -8\\ -4& -18& -48\end{array}\right|=0\Rightarrow \left|\begin{array}{ccc}x-2& 2x-3& 3x-4\\ 1& 3& 4\\ 2& 9& 24\end{array}\right|=0$

Expand along the first row
$(x-2)\left(36\right)-(2x-3)\left(16\right)+(3x-4)\left(3\right)=0\Rightarrow 13x-28=0\Rightarrow x=28/13$