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Section 3.2 : Lines
8. Determine if the two lines\(y = \frac{3}{7}x + 1\) and \(3y + 7x = - 10\) are parallel, perpendicular or neither.
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Start SolutionTo answer this question we’ll need the slope of each of the lines. The first line is in slope-intercept form and so we can easily identify the slope of that line.
\[y = \frac{3}{7}x + 1\,\,\,\hspace{0.25in}:\hspace{0.25in}{m_1} = \frac{3}{7}\]For the second line let’s put the equation in slope-intercept form and get its slope.
\[\begin{align*}3y + 7x & = - 10\\ 3y & = - 7x - 10\\ y & = - \frac{7}{3}x - \frac{{10}}{3}\hspace{0.25in}:\hspace{0.25in}{m_2} = - \frac{7}{3}\end{align*}\] Show Step 2The two slopes we found in the previous step are clearly not the same and so the two lines are not parallel.
On the other hand, we can see that,
\[{m_1}{m_2} = \left( {\frac{3}{7}} \right)\left( { - \frac{7}{3}} \right) = - 1\]and so the two lines are perpendicular.