Question:
Find the value of $x$ for which the points $A(x, 2), B(-3,-4)$ and $C(7,-5)$ are collinear.
Solution:
Let $A\left(x_{1}, y_{1}\right)=A(x, 2), B\left(x_{2}, y_{2}\right)=B(-3,-4)$ and $C\left(x_{3}, y_{3}\right)=C(7,-5)$. So, the condition for three collinear points is
$x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)=0$
$\Rightarrow x(-4+5)-3(-5-2)+7(2+4)=0$
$\Rightarrow x+21+42=0$
$\Rightarrow x=-63$
Hence, x = − 63.