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Section 3.2 : Lines
9. Determine if the line \(8x - y = 2\) and the line containing the two points \(\left( {1,3} \right)\) and \(\left( {2, - 4} \right)\) are parallel, perpendicular or neither.
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Start SolutionTo answer this question we’ll need the slope of each of the lines. For the first line let’s put the equation in slope-intercept form and get its slope.
\[\begin{align*}8x - y & = 2\\ y & = 8x - 2\hspace{0.25in}:\hspace{0.25in}{m_1} = 8\end{align*}\]For the second line we can compute the slope directly from the two points.
\[{m_2} = \frac{{ - 4 - 3}}{{2 - 1}} = \frac{{ - 7}}{1} = - 7\] Show Step 2The two slopes we found in the previous step are clearly not the same and so the two lines are not parallel. Also, we can see that \({m_1}{m_2} = - 56 \ne - 1\) and so the lines are also not perpendicular.
Therefore, the two lines are neither parallel nor perpendicular.