Question

How do you tell whether the lines for each pair of equations are parallel, perpendicular, or neither: $y = - 4x + 3,{\text{ }} - 2x + 8y = 5$?

Answer 1

Hint: Two lines will be parallel if their slopes are equal. Also, two lines are perpendicular if the product of their slopes is -1. To determine the slope of the equation write it in its slope-intercept form.

Complete step by step solution:
We are supposed to determine whether the two lines, whose equations are given, are parallel, perpendicular or neither parallel nor perpendicular. We know that two lines would be parallel if their slopes are equal and would be perpendicular if the product of their slopes is equal to -1. So, we will use these results to solve this question.
Firstly, we will find the slopes of the given lines. For this we will write the equation in the slope-intercept form $y = mx + c$. And $m$ in this equation gives the slope of the line.
We have the equation of the lines as,
$y = - 4x + 3$ - - - - - - - - - - - (1)
$ - 2x + 8y = 5$ - - - - - - - - - - - (2)
To write the equations in the slope intercept form, first bring the y term to one side and other terms to the other side. Then from (1) and (2) we get,
$ \Rightarrow y = - 4x + 3$ and $8y = 2x + 5$
Then, divide the entire equation by the coefficient of the y term.
$ \Rightarrow y = - 4x + 3$ and $y = \dfrac{1}{4}x + \dfrac{5}{8}$
Now, comparing these equations with the slope intercept form $y = mx + c$, we get the slopes of the slope of equation (1) as $ - 4$ and the slope of equation (2) as $\dfrac{1}{4}$. Now we see that their slopes are not equal. So, they are not parallel lines.
Now, their product $ - 4 \times \dfrac{1}{4} = - 1$. So, the product of these two equations slopes is -1.
Hence the lines $y = - 4x + 3,{\text{ }} - 2x + 8y = 5$ are perpendicular.

Note: Always remember to not make any mistakes while representing the equation in the slope-intercept form $y = mx + c$. Also, here $c$ gives the $y$ intercept of the line.