Question

If the points A(x, y), B(−5, 7) and C(−4, 5) are collinear, show that 2x + y + 3 = 0.

Answer 1

The given points are $A\left(x,y\right),B\left(-5,7\right)\text{and}C\left(-4,5\right)$.
Here, $\left(x_1=x,y_1=y\right),\left(x_2=-5,y_2=7\right)\text{and}\left(x_3=-4,y_3=5\right)$
A,B and C​ are collinear. Then,
$$\begin{gathered} x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)=0 \\ \Rightarrow x\left(7-5\right)-5\left(5-y\right)-4\left(y-7\right)=0 \\ \Rightarrow 7x-5x-25+5y-4y+28=0 \\ \Rightarrow 2x+y+3=0 \end{gathered}$$