What is the slope of a line perpendicular to -x - 2y = 8?
Understand the Problem
The question is asking for the slope of a line that is perpendicular to the line represented by the equation -x - 2y = 8. To solve this, we first need to find the slope of the given line and then determine the negative reciprocal of that slope, since the slopes of perpendicular lines are negative reciprocals of each other.
Answer
$2$
Answer for screen readers
The slope of a line perpendicular to $-x - 2y = 8$ is $2$.
Steps to Solve
- Convert the equation to slope-intercept form
To find the slope of the line represented by the equation $-x - 2y = 8$, we need to convert it to the slope-intercept form, $y = mx + b$, where $m$ is the slope.
Starting with: $$ -x - 2y = 8 $$
We can isolate $y$: $$ -2y = x + 8 $$
Now divide by -2: $$ y = -\frac{1}{2}x - 4 $$
Thus, the slope ($m$) of the original line is $-\frac{1}{2}$.
- Determine the negative reciprocal of the slope
Since the slope of the original line is $-\frac{1}{2}$, we need to find the negative reciprocal to get the slope of the perpendicular line.
The negative reciprocal of $-\frac{1}{2}$ is: $$ m_{\text{perpendicular}} = -\left(-\frac{1}{2}\right)^{-1} = 2 $$
- Final result for the slope
So, the slope of a line perpendicular to the given line is: $$ m_{\text{perpendicular}} = 2 $$
The slope of a line perpendicular to $-x - 2y = 8$ is $2$.
More Information
The concept of perpendicular slopes is based on the fact that perpendicular lines have slopes that are negative reciprocals of each other. This means if one line has a slope of $m$, the perpendicular line will have a slope of $-\frac{1}{m}$.
Tips
- Confusing the negative reciprocal: It's easy to forget that the reciprocal should also change its sign.
- Not converting to slope-intercept form: Some might attempt to read the slope directly from a standard form equation without rearranging it first.
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