Understanding Perpendicular Lines and Their Slopes

For any line with the equation y mx b, where m is the slope, the line that is perpendicular to it will have a slope that is the negative reciprocal of m. This information forms the basis for understanding the relationships between lines in coordinate geometry.

The Problem at Hand

We are tasked with finding the equation of a line that passes through the point (2, -1) and is perpendicular to the line given by the equation y -5x 1. This involves several steps, including identifying the slope of the given line, finding the slope of the perpendicular line, and using the point-slope form to find the equation of the new line.

Step 1: Identify the Slope of the Given Line

The given line has the equation y -5x 1. From this equation, we can identify the slope of the given line as -5.

Step 2: Determine the Slope of the Perpendicular Line

The slope of a line perpendicular to y -5x 1 can be found using the formula for the negative reciprocal. Thus, the slope of the required line is the negative reciprocal of -5, which is 1/5.

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

The point-slope form of a line is given by the equation y - y1 m(x - x1), where (x1, y1) is a point on the line and m is the slope. Substituting the point (2, -1) and the slope 1/5, we get:

y - (-1) 1/5 (x - 2)

This simplifies to:

y 1 1/5 (x - 2)

Multiplying through by 5 to eliminate the fraction:

5(y 1) x - 2

Simplifying, we get:

x - 5y - 7 0

Step 4: Convert to Standard Form

The standard form of a linear equation is Ax By C. We can rearrange the equation obtained in Step 3 to the standard form:

x - 5y 7

Conclusion

Thus, the equation of the line that is perpendicular to y -5x 1 and passes through the point (2, -1) is x - 5y 7.

Understanding how to find the equation of a line perpendicular to a given line involves recognizing the relationship between the slopes of the lines and using the point-slope form to derive the equation of the new line.