Find the value of x if the area
Question:

Find the value of $x$ if the area of $\Delta$ is 35 square $\mathrm{cms}$ with vertices $(x, 4),(2,-6)$ and $(5,4)$.

Solution:

$\Delta=\frac{1}{2}\left|\begin{array}{ccc}x & 4 & 1 \\ 2 & -6 & 1 \\ 5 & 4 & 1\end{array}\right|=\pm 35$

$=\frac{1}{2}\left|\begin{array}{ccc}x & 4 & 1 \\ 2-x & -10 & 0 \\ 5 & 4 & 1\end{array}\right|=\pm 35 \quad\left[\right.$ Applying $\left.R_{2} \rightarrow R_{2}-R_{1}\right]$

$=\frac{1}{2}\left|\begin{array}{ccc}x & 4 & 1 \\ 2-x & -10 & 0 \\ 5-x & 0 & 0\end{array}\right|=\pm 35 \quad$ [Applying $R_{3} \rightarrow R_{3}-R_{1}$ ]

$=\frac{1}{2}\left|\begin{array}{cc}2-x & -10 \\ 5-x & 0\end{array}\right|=\pm 35$

$=0+10(5-x)=\pm 70$

$\Rightarrow 50-10 x=70$ or $50-10 x=-70$

$\Rightarrow-10 x=20$ or $-10 x=-120$

$\Rightarrow x=-2$ or $x=12$