Need help to find the perpendicular of y = 1/3x + 5
Understand the Problem
The question is asking how to find the equation of a line that is perpendicular to another line given by the equation y = (1/3)x + 5. To find this, we need to identify the slope of the original line and then use the negative reciprocal of that slope for the perpendicular line.
Answer
The equation of the perpendicular line is $y = -3x$.
Answer for screen readers
The equation of the line that is perpendicular to the line $y = \frac{1}{3}x + 5$ which passes through the origin is $y = -3x$.
Steps to Solve
- Identify the slope of the original line
The given equation of the line is in the slope-intercept form $y = mx + b$, where $m$ is the slope. Here, the slope $m$ of the original line is $\frac{1}{3}$.
- Find the negative reciprocal of the slope
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. To find the negative reciprocal, take the negative of the slope and then flip it:
$$ \text{Perpendicular slope} = -\frac{1}{\frac{1}{3}} = -3 $$
- Write the equation of the perpendicular line
Using the slope-intercept form again, we can write the equation of the line with the slope $-3$. This needs a point through which it passes (not stated in the question, so we typically use some arbitrary point like (0,0) or you could define one). Assuming we want it to pass through the point (0,0), the equation becomes:
$$ y = -3x $$
If you have a specific point in mind, you can substitute that point into the equation $y = mx + b$ to find $b$ and adjust accordingly.
The equation of the line that is perpendicular to the line $y = \frac{1}{3}x + 5$ which passes through the origin is $y = -3x$.
More Information
The negative reciprocal of a slope is a crucial concept when dealing with perpendicular lines in geometry. When you know the slope of one line, you can always find the slope of the perpendicular line.
Tips
- Forgetting to take the negative of the slope when calculating the negative reciprocal.
- Not recognizing that the slope of a vertical line is undefined, while the slope of a horizontal line is zero, leading to possible confusion when dealing with perpendicularity.
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