Understanding Parallel Lines and Their Equations
Parallel lines are a fundamental concept in geometry, defined by their property of having the same slope. Understanding how to find the equation of a line parallel to a given line and passing through a specific point is crucial for various applications in mathematics and engineering. This guide will walk you through the process step-by-step.
Key Concepts and Formulas
Key Concepts:
Parallel lines have the same slope. The slope-intercept form of a line is (y mx b), where (m) is the slope and (b) is the y-intercept. The point-slope form of a line is (y - y_1 m(x - x_1)), where (m) is the slope and ((x_1, y_1)) is a point on the line. A graphical solution involves rewriting the line equation in the form (ax by c).Step-by-Step Solution
Given the line 4x - 8y 1, we need to find the equation of a line parallel to it and passing through the point (2, -8).
Step 1: Convert the Given Line to Slope-Intercept Form
The given line is (4x - 8y 1). To find the slope, we first need to convert it to the slope-intercept form (y mx b).
[4x - 8y 1]
Rearrange to solve for (y):
[-8y -4x 1]
[y frac{1}{2}x - frac{1}{8}]
From this, we can see that the slope (m frac{1}{2}).
Step 2: Use the Point-Slope Form with the Given Point
Since the parallel line has the same slope, its slope is also (m frac{1}{2}). We use the point-slope form (y - y_1 m(x - x_1)) with the point ((2, -8)).
[y - (-8) frac{1}{2}(x - 2)]
[y 8 frac{1}{2}x - 1]
[y frac{1}{2}x - 9]
Therefore, the equation of the line parallel to (4x - 8y 1) and passing through the point ((2, -8)) is (y frac{1}{2}x - 9).
Alternative Methods
Graphical Solution:
We can also consider the line in the form (4x - 8y a). To find (a), we use the point ((2, -8)).
[4(2) - 8(-8) a]
[8 64 a]
[a 72]
Thus, the equation of the line is (4x - 8y 72).
Conclusion
By understanding the properties of parallel lines and using appropriate forms of linear equations, we can easily find the equation of a line parallel to a given line and passing through a specific point.