How to make an equation perpendicular?
Understand the Problem
The question is asking how to modify an equation so that it represents a line that is perpendicular to another line. The high-level approach involves understanding the slope of the original line and then using the negative reciprocal of that slope for the perpendicular line.
Answer
The equation of the perpendicular line is: $$ y - y_1 = -\frac{1}{m}(x - x_1) $$
Answer for screen readers
The equation of the line that is perpendicular to the original line is given by:
$$ y - y_1 = -\frac{1}{m}(x - x_1) $$
Steps to Solve
- Identify the slope of the original line
First, locate the equation of the original line. If it’s in the form $y = mx + b$, where $m$ represents the slope, read the value of $m$. This value will be used to find the slope of the perpendicular line.
- Calculate the slope of the perpendicular line
The slope of the perpendicular line can be found by taking the negative reciprocal of the original line's slope. If the slope of the original line is $m$, the slope of the perpendicular line will be $m_{perpendicular} = -\frac{1}{m}$.
- Write the equation for the perpendicular line
Now, use the point-slope form of a line to write the equation. If the perpendicular line passes through a specific point $(x_1, y_1)$, the equation will be:
$$ y - y_1 = m_{perpendicular}(x - x_1) $$
Substituting $m_{perpendicular}$ from the previous step gives:
$$ y - y_1 = -\frac{1}{m}(x - x_1) $$
- Simplify the equation of the perpendicular line
After expanding and rearranging, you might want to simplify your final equation into slope-intercept form $y = mx + b$ if needed.
The equation of the line that is perpendicular to the original line is given by:
$$ y - y_1 = -\frac{1}{m}(x - x_1) $$
More Information
When dealing with slopes, remember that perpendicular lines have slopes that multiply to $-1$. This geometric property is key in many areas of math, from coordinate geometry to calculus.
Tips
- Forgetting to take the negative reciprocal of the slope, which leads to an incorrect slope for the perpendicular line.
- Not properly substituting the point $(x_1, y_1)$ into the equation, which can result in an equation not depicting the correct line.
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