Determining the Equation of a Perpendicular Line with a Given Y-Intercept

This article walks you through the step-by-step process of determining the equation of a line that is perpendicular to a given line and also has a specified y-intercept. The process involves calculating the slope of the original line, finding the slope of the perpendicular line, and then using the slope-intercept form to find the final equation.

Understanding the Problem

The problem at hand is to find the equation of a line that is perpendicular to the line passing through the points (-1, 2) and (3, 1) and has a y-intercept of 2.

Step 1: Calculate the Slope of the Given Line

The slope of the line passing through the points (-1, 2) and (3, 1) can be calculated using the formula for the slope:

slope  (y2 - y1) / (x2 - x1)

Substituting the given points:

slope  (1 - 2) / (3 - -1)  -1 / 4

The slope of the given line is -1/4.

Step 2: Determine the Slope of the Perpendicular Line

The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope of the perpendicular line is:

slope of perpendicular line  -1 / (-1/4)  4

The slope of the perpendicular line is 4.

Step 3: Use the Slope-Intercept Form to Find the Equation

The slope-intercept form of a line is given by:

Y  mX   c

Where m is the slope of the line and c is the y-intercept. For the perpendicular line, we know the y-intercept is 2 and the slope is 4. Therefore, the equation of the perpendicular line is:

y  4x   2

By following these steps, we can accurately determine the equation of a line that is perpendicular to a given line and has a specified y-intercept.

Conclusion

To summarize, determining the equation of a perpendicular line with a given y-intercept involves calculating the slope of the original line, finding the slope of the perpendicular line, and then using the slope-intercept form. This method provides a clear and systematic approach to solving such problems in geometry and algebra.

Understanding these concepts is crucial in many areas of mathematics, including calculus and engineering. By mastering these fundamentals, students and professionals alike can confidently tackle more complex problems in their respective fields.

If you have any further questions or need additional clarification, feel free to ask. Happy learning!