What Is The Equation, In Slope-intercept Form, Of The Line That Is Perpendicular To The Given Line And Passes Through The Point $(2, -1)$?A. $y = -\frac{1}{3} X - \frac{1}{3}$ B. $y = -\frac{1}{3} X - \frac{5}{3}$ C.

Feb 28, 2025 by ADMIN 221 views

Introduction

In mathematics, the slope-intercept form of a line is a fundamental concept that is used to represent the equation of a line in a unique and simplified way. The slope-intercept form is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. In this article, we will explore the concept of finding the equation of a line that is perpendicular to a given line and passes through a specific point.

Understanding the Concept of Perpendicular Lines

Perpendicular lines are lines that intersect at a 90-degree angle. In other words, if two lines are perpendicular, they form a right angle at the point of intersection. The slope of a line is a measure of how steep it is, and the slope of a perpendicular line is the negative reciprocal of the slope of the original line. This means that if the slope of the original line is m, the slope of the perpendicular line is -1/m.

Finding the Equation of a Perpendicular Line

To find the equation of a line that is perpendicular to a given line and passes through a specific point, we need to follow these steps:

Find the slope of the given line.
Find the negative reciprocal of the slope of the given line, which is the slope of the perpendicular line.
Use the point-slope form of a line to find the equation of the perpendicular line.
Simplify the equation to find the slope-intercept form.

Step 1: Find the Slope of the Given Line

The given line is not explicitly mentioned in the problem, but we can assume that it is a line with a slope of m. To find the slope of the given line, we need to know the equation of the line. However, since the equation of the line is not given, we can assume that the slope of the given line is m.

Step 2: Find the Negative Reciprocal of the Slope of the Given Line

The negative reciprocal of the slope of the given line is -1/m. This is the slope of the perpendicular line.

Step 3: Use the Point-Slope Form of a Line to Find the Equation of the Perpendicular Line

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 of the line. In this case, the point (2, -1) is on the perpendicular line, and the slope of the perpendicular line is -1/m. Substituting these values into the point-slope form, we get:

y - (-1) = (-1/m)(x - 2)

Step 4: Simplify the Equation to Find the Slope-Intercept Form

To simplify the equation, we can start by distributing the slope to the terms inside the parentheses:

y + 1 = (-1/m)(x - 2)

Next, we can multiply both sides of the equation by m to eliminate the fraction:

my + m = -x + 2

Now, we can rearrange the terms to get the equation in slope-intercept form:

my = -x + 2 - m

y = (-1/m)x + (2 - m)/m

Step 5: Find the Value of m

Since the slope of the given line is m, we can substitute this value into the equation:

y = (-1/m)x + (2 - m)/m

However, we are not given the value of m. To find the value of m, we need to know the equation of the given line. Since the equation of the given line is not given, we can assume that the value of m is not relevant to the problem.

Step 6: Find the Equation of the Perpendicular Line

Since the value of m is not relevant to the problem, we can assume that the equation of the perpendicular line is:

y = (-1/m)x + (2 - m)/m

However, this equation is not in the correct form. To find the correct equation, we need to substitute the value of m into the equation. Since the value of m is not given, we can assume that the equation of the perpendicular line is:

y = (-1/m)x + (2 - m)/m

However, this equation is not in the correct form. To find the correct equation, we need to simplify the equation further.

Step 7: Simplify the Equation Further

To simplify the equation further, we can start by multiplying both sides of the equation by m:

my = (-1/m^2)x + (2 - m)/m

Next, we can multiply both sides of the equation by m^2 to eliminate the fraction:

m^2y = -x + 2m - m^2

Now, we can rearrange the terms to get the equation in slope-intercept form:

m^2y = -x + 2m - m^2

y = (-1/m^2)x + (2m - m^2)/m2

Step 8: Simplify the Equation Further

To simplify the equation further, we can start by factoring out the common term m^2:

y = (-1/m^2)x + (2m - m^2)/m2

Next, we can simplify the numerator of the second term:

y = (-1/m^2)x + (2m - m^2)/m2

y = (-1/m^2)x + (2m/m^2 - m^2/m2)

y = (-1/m^2)x + (2/m - 1)