Question

For what value of x are the points A(−3, 12), B(7, 6) and C(x, 9) collinear?

Answer 1

A(−3, 12), B(7, 6) and C(x, 9) are the given points. Then:
(x₁ = −3, y₁ = 12), (x₂ = 7, y₂ = 6) and (x₃ = x, y₃ = 9)
It is given that the points A, B and C are collinear. Therefore,
$$\begin{gathered} x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)=0 \\ \Rightarrow \left(-3\right)\left(6-9\right)+7\left(9-12\right)+x\left(12-6\right)=0 \\ \Rightarrow \left(-3\right)\left(-3\right)+7\left(-3\right)+x\left(6\right)=0 \\ \Rightarrow 9-21+6x=0 \\ \Rightarrow 6x-12=0 \\ \Rightarrow 6x=12 \\ \Rightarrow x=\frac{12}{6}=2 \end{gathered}$$
Therefore, when x= 2, the given points are collinear.