Introduction to Finding the Equation of a Parallel Line
In this article, we delve into the process of finding the equation of a line that is parallel to another line and passes through a specific point. We'll explore the math involved and provide step-by-step instructions to help you solve similar problems effectively.
Understanding Parallel Lines and Their Slopes
Parallel lines in geometry are lines that never intersect and maintain a constant distance from each other. One of the key characteristics of parallel lines is that they have the same slope. If you have a line with the equation y - 1 4x 3, the slope (m) of this line is 4. Consequently, any line parallel to this one will also have a slope of 4.
Deriving the Equation of the Parallel Line
The equation of a line can be written in the point-slope form: y - y_1 m(x - x_1). Here, (x_1, y_1) is a point on the line, and m is the slope of the line.
Step-by-Step Derivation
Identify the slope from the given line equation: y - 1 4x 3. The slope (m) is 4. Use the point-slope form equation with the point (4, 32) and the slope 4:y - 32 4(x - 4)Expand and simplify the equation:
y - 32 4x - 16or equivalently,
y 4x 16
This equation represents the line parallel to y - 1 4x 3 that passes through the point (4, 32).
Alternative Method: Simplifying and Using the Given Point
Alternatively, you can simplify the initial line equation y - 1 4x 3 to make the process easier. By simplifying, we get:
y 4x 2Since parallel lines have the same slope, the equation of any parallel line will also have the form y 4x c. To find the specific value of c for the line passing through (4, 32), substitute the point (4, 32) into the equation:
32 4(4) c 32 16 c c 16
Therefore, the equation of the parallel line is:
y 4x 16
Conclusion
By understanding the fundamental principles of parallel lines and the point-slope form of a line's equation, you can easily find the equation of a line that is parallel to another line and passes through a specific point. Whether you follow the step-by-step derivation or simplify the process, the key is to maintain the same slope and use the given point to find the correct y-intercept.