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Chapter 1: Problem 71

Determine whether the lines \(L_{1}\) and \(L_{2}\) passing through the pairs of points are parallel, perpendicular, or neither. $$\begin{aligned} &L_{1}:(3,6),(-6,0)\\\ &L_{2}:(0,-1),\left(5, \frac{7}{3}\right) \end{aligned}$$

Short Answer

Expert verified
The lines \(L_{1}\) and \(L_{2}\) are parallel.

Step by step solution

01

Compute the Slope of \(L_{1}\)

We can calculate the slope of a line using two points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) with the formula \(m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\). Therefore, for the line \(L_{1}\) passing through the points (3,6) and (-6,0), the slope \(m_{1}\) is: \[m_{1} = \frac{0-6}{-6-3}=\frac{-6}{-9}=\frac{2}{3}\]
02

Compute the Slope of \(L_{2}\)

Similarly, for the line \(L_{2}\) passing through the points \((0,-1)\) and \(\left(5, \frac{7}{3}\right)\), the slope \(m_{2}\) is: \[m_{2} = \frac{\frac{7}{3}-(-1)}{5-0}=\frac{\frac{10}{3}}{5}=\frac{2}{3}\]
03

Analyze the Relationship between the Lines

The lines \(L_{1}\) and \(L_{2}\) are parallel if their slopes are equal. By comparing the slopes \(m_{1}\) and \(m_{2}\) computed in steps 1 and 2, we find \(m_{1} = m_{2}\) which means the lines \(L_{1}\) and \(L_{2}\) are parallel.

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