Finding the Equation of a Perpendicular Line
When dealing with lines in a two-dimensional plane, understanding the properties of perpendicular lines is crucial. In this article, we will explore the process of finding the equation of a line that is perpendicular to a given line and passes through a specific point. We will delve into the mathematical concepts, step-by-step solutions, and explain the underlying principles in a clear and concise manner.
Understanding the Problem
The problem at hand is to find the equation of a line that is perpendicular to the line given by the equation 2y x 5 and that passes through the point (-14, 0).
Step 1: Simplify the Given Line Equation
The first step is to simplify the given line equation 2y x 5. By dividing both sides by 2, we can express this equation in slope-intercept form y mx b, where m is the gradient.
2y x 5 y 1/2 x 5/2
In this equation, the gradient m is 1/2. This slope represents the rate of change of the line, indicating how much the line rises or falls for a given horizontal distance.
Step 2: Determine the Gradient of the Perpendicular Line
Lines that are perpendicular to each other have gradients that are negative reciprocals of each other. Therefore, if the gradient of the first line is 1/2, the gradient of the perpendicular line will be the negative reciprocal of 1/2, which is -2.
The formula for the negative reciprocal is given by:
m_perpendicular -1 / m_original
Step 3: Use the Point-Slope Form to Find the Perpendicular Line Equation
Now, we need to find the equation of the line that has a gradient of -2 and passes through the point (-14, 0). The point-slope form of a line equation is given by:
y - y1 m(x - x1)
Where (x1, y1) is the point through which the line passes. In this case, x1 -14 and y1 0.
Step 4: Substitute the Known Values
Substitute the values -2 for the gradient m, and (-14, 0) for the point into the point-slope form equation:
y - 0 -2(x - (-14))
Which simplifies to:
y -2(x 14)
Step 5: Simplify the Equation
Expand and simplify the equation:
y -2x - 28
Thus, the equation of the line that is perpendicular to 2y x 5 and passes through the point (-14, 0) is:
y -2x - 28
Conclusion
This process demonstrates the step-by-step method of finding the equation of a perpendicular line. By understanding the gradient of the original line and the properties of perpendicular lines, we can determine the equation of the desired line that meets specific conditions.
Further Reading and Practice
For a deeper understanding of the concepts involved, explore additional resources on algebraic geometry and linear equations. Practicing similar problems will help reinforce your skills and build confidence in solving such mathematical challenges.