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Question:

If \begin{vmatrix} 3x & -7 \ -2 & 4 \end{vmatrix} = \begin{vmatrix} 8 & -7 \ 6 & 4 \end{vmatrix}, find the value of x.

Solution:

Compute the determinants as shown below:
\begin{vmatrix} 3x & -7 \ -2 & 4 \end{vmatrix} = \begin{vmatrix} 8 & -7 \ 6 & 4 \end{vmatrix}
(3x)(4) - (-2)(-7) = 8 \times 4 - 6 \times (-7)
\Rightarrow 12x + 14 = 32 + 42
\Rightarrow 12x = 74 - 14
\Rightarrow 12x = 60
\Rightarrow x = 5