Q 4.103 - Intermediate Algebra

Question

Determine whether the points (3, −2), (5, −3), and (1, −1) are collinear.

Step-by-Step Solution

Answer

The  Given points are  collinear. 

Step 1. Given information

The given points are  (3, −2), (5, −3), and (1, −1)  .

Step 2. Calculation

$$\begin{gathered} \left[\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right] \\ \mathrm{Substitute} \mathrm{the} \mathrm{values} \mathrm{into} \mathrm{the} \mathrm{determinant}. \\ (3, -2), (5, -3), \mathrm{and} (1, -1) . \\ D=\left[\begin{array}{ccc}3& -2& 1\\ 5& -3& 1\\ 1& -1& 1\end{array}\right] \\ \mathrm{Evaluate} \mathrm{the} \mathrm{determinant} \mathrm{by} \mathrm{expanding} \mathrm{by} \mathrm{minors} \mathrm{using} \mathrm{column} 3. \\ 1\left[\begin{array}{cc}5& -3\\ 1& -1\end{array}\right]-1\left[\begin{array}{cc}3& -2\\ 1& -1\end{array}\right]+1\left[\begin{array}{cc}3& -2\\ 5& -3\end{array}\right] \\ =(-5+3)-(-3+2)+(-9+10) \\ =-2+1+1 \\ =0 \\ \mathrm{The} \mathrm{value} \mathrm{of} \mathrm{the} \mathrm{determinate} \mathrm{is} 0, \mathrm{so} \mathrm{the} \mathrm{points} \mathrm{are} \mathrm{collinear}. \end{gathered}$$