Q 4.104 - Intermediate Algebra

Question

 Determine whether the points (−4, −1), (−6, 2), and (−2, −4) are collinear.

Step-by-Step Solution

Answer

The  Given points are  collinear.  

Step 1. Given information

The given points are  (−4, −1), (−6, 2), and (−2, −4)  .

Step 2. Calculation

$$\begin{gathered} \left[\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right] \\ \mathrm{Substitute} \mathrm{the} \mathrm{values} \mathrm{into} \mathrm{the} \mathrm{determinant}. \\ (-4, -1), (-6, 2), \mathrm{and} (-2, -4) . \\ D=\left[\begin{array}{ccc}-4& -1& 1\\ -6& 2& 1\\ -2& -4& 1\end{array}\right] \\ \mathrm{Evaluate} \mathrm{the} \mathrm{determinant} \mathrm{by} \mathrm{expanding} \mathrm{by} \mathrm{minors} \mathrm{using} \mathrm{column} 3. \\ 1\left[\begin{array}{cc}-6& 2\\ -2& -4\end{array}\right]-1\left[\begin{array}{cc}-4& -1\\ -2& -4\end{array}\right]+1\left[\begin{array}{cc}-4& -1\\ -6& 2\end{array}\right] \\ =(24+4)-(16-2)+(-8-6) \\ =28-14-14 \\ =0 \\ \mathrm{The} \mathrm{value} \mathrm{of} \mathrm{the} \mathrm{determinate} \mathrm{is} 0, \mathrm{so} \mathrm{the} \mathrm{points} \mathrm{are} \mathrm{collinear}. \end{gathered}$$