If the points (x, −2), (5, 2), (8, 8) are collinear, find x using determinants.

If the points (x, −2), (5, 2), (8, 8) are collinear, then

$\left|\begin{array}{ccc}x & -2 & 1 \\ 5 & 2 & 1 \\ 8 & 8 & 1\end{array}\right|=0$

$\Delta=\left|\begin{array}{ccc}x & -2 & 1 \\ 5 & 2 & 1 \\ 8 & 8 & 1\end{array}\right|$

$\Delta=\left|\begin{array}{ccc}x & -2 & 1 \\ 5-x & 4 & 0 \\ 8 & 8 & 1\end{array}\right| \quad\left[\right.$ Applying $\left.R_{2} \rightarrow R_{2}-R_{1}\right]$

$=\left|\begin{array}{ccc}x & -2 & 1 \\ 5-x & 4 & 0 \\ 8-x & 10 & 0\end{array}\right| \quad\left[\right.$ Applying $\left.R_{3} \rightarrow R_{3}-R_{1}\right]$

$=\left|\begin{array}{cc}5-x & 4 \\ 8-x & 10\end{array}\right|$

$=50-10 x-32+4 x$

$=18-6 x$

$\Delta=18-6 x$

$\Delta=0 \quad$ [Given]

$\Rightarrow 18-6 x=0$

$\Rightarrow x=3$