Question

Find the value of x for which the points A(x, 2), B(-3, -4) and C(7, -5) are collinear.

Answer 1

Let $A(x_1,y_1)=A(x,2)$,
$B(x_2,y_2)=B(-3,-4)$ and
$C(x_3,y_3)=C(7,-5)$.

So the condition for three collinear points is the area is made by these points is zero

$Area=\frac{1}{2}\left[x_1\right(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2\left)\right]$

$0=\frac{1}{2}\left[x\right(-4+5)-3(-5–2)+7(2+4\left)\right]$

$0=\frac{1}{2}[x+21+42]$

$x=–63$