Parallel lines
Watch this video to learn about parallel lines.
Two lines have equations \(y = {m_1}x + 3\) and \(y = {m_2}x - 7\).
If the lines are parallel then \({m_1} = {m_2}\) and if \({m_1} = {m_2}\) then the lines are parallel.
Example
Show that the lines with equations \(2y = x + 1\) and \(3x - 6y - 8 = 0\) are parallel.
Solution
First rearrange each equation in the form \(y = mx + c\).
\(2y = x + 1\)
\(y = \frac{1}{2}x + \frac{1}{2}\)
Identify the first gradient:
\(gradient = \frac{1}{2}\)
\(3x - 6y - 8 = 0\)
\(- 6y = - 3x + 8\)
\(y = \frac{{ - 3x}}{{ - 6}} + \frac{8}{{ - 6}}\)
\(y = \frac{1}{2}x - \frac{4}{3}\)
Identify the second gradient:
\(gradient = \frac{1}{2}\)
Complete the proof:
Gradients are equal, hence lines are parallel.
Equations of lines where the gradient is undefined
The gradient of a line which is parallel to the \(y\)-axis cannot be determined from the above formulae. Just write the equation down!
Lines parallel to y-axis
In the same way, the equations of lines which are parallel to the \(x\)-axis can also be written down.
