Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.
Section 3.2 : Lines
11. Find the equation of the line through \(\left( { - 7,2} \right)\) and is perpendicular to the line \(3x - 14y = 4\).
Show All Steps Hide All Steps
Start SolutionFirst, we need to get the slope of our new line, i.e. the line through the point \(\left( { - 7,2} \right)\). We know this line is parallel to the line \(3x - 14y = 4\) and the slopes must be negative reciprocals of each other.
Therefore, all we need to do is put the equation of the second line into slope-intercept form and get its slope.
\[\begin{align*}3x - 14y & = 4\\ 14y & = 3x - 4\\ y & = \frac{3}{{14}}x - \frac{2}{7}\hspace{0.25in}:\hspace{0.25in}{m_2} = \frac{3}{{14}}\end{align*}\]So, the new line must have slope of,
\[{m_1} = - \frac{1}{{{m_2}}} = - \frac{1}{{{}^{3}/{}_{{14}}}} = - \frac{{14}}{3}\] Show Step 2Now, we have both the slope of the new line as well as a point through the new line so we can use the point-slope form of the line to write down the equation of the new line.
\[y = 2 - \frac{{14}}{3}\left( {x - \left( { - 7} \right)} \right) = 2 - \frac{{14}}{3}\left( {x + 7} \right) = - \frac{{14}}{3}x - \frac{{92}}{3}\]