Understanding Perpendicular Lines and Their Equations
Perpendicular lines are two lines that intersect at a right angle. Understanding the relationship between their slopes is crucial for finding the equation of a line that is perpendicular to another line and passes through a specific point. This article will walk you through the process step-by-step.
Step-by-Step Guide to Finding the Perpendicular Line's Equation
Given a line with the equation y 3x - 2, we want to find the equation of a line that is perpendicular to it and passes through the point (2, 3).
Identify the Slope of the Given Line
The slope of the given line can be directly identified from its equation. For the line y 3x - 2, the slope m1 is 3.
Determine the Slope of the Perpendicular Line
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Therefore, the slope m2 of the perpendicular line is:
m2 -1/m1 -1/3
Use the Point-Slope Form of the Equation
The point-slope form of the equation of a line is given by:
y - y1 m(x - x1)
Here, x1 and y1 are the coordinates of a point on the line, and m is the slope. For the given point (2, 3) and slope -1/3, we can substitute these values into the equation:
y - 3 -1/3(x - 2)
Simplify the Equation
To convert the equation into its slope-intercept form y mx b, follow these steps:
y - 3 -1/3x 2/3
y -1/3x 2/3 3
y -1/3x 2/3 9/3
y -1/3x 11/3
Alternative Standard Form
If you prefer the standard form Ax By C, the equation can be rearranged as:
x 3y 11
Explanation of the Solution
By following the steps above, we can find the equation of the line that is perpendicular to the given line y 3x - 2 and passes through the point (2, 3). This process helps us understand the relationship between the slopes of perpendicular lines and how to use the point-slope form to find the desired line's equation.
Conclusion
Understanding and solving problems involving perpendicular lines and their equations can be crucial for various applications in mathematics and real-world scenarios. Whether you need to find the equation in slope-intercept form or the standard form, the steps outlined in this article provide a clear and effective method.