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Chapter 3: Problem 80

Find the equation of the line in slope-intercept form. Perpendicular to \(y=1\) and passing through (4,-1) .

Short Answer

Expert verified
The equation of the line is \( x = 4 \).

Step by step solution

01

Identify Slope of Given Line

The equation given is \( y = 1 \), which is a horizontal line. Horizontal lines have a slope of 0.
02

Determine Slope of Perpendicular Line

Perpendicular lines have slopes that are negative reciprocals of each other. For a line with a slope of 0, the perpendicular slope is undefined, indicating a vertical line.
03

Recognize Characteristics of Vertical Line

Vertical lines have equations of the form \( x = a \). The slope-intercept form \( y = mx + b \) is not applicable here, but we can represent the line as \( x = a \).
04

Equation of the Line Passing Through (4,-1)

Since the line is vertical and passes through the point \((4, -1)\), the equation of the line is simply \( x = 4 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perpendicular Lines
Perpendicular lines are a fascinating concept in geometry. They intersect at a 90-degree angle, creating distinct features in their equations.
  • Understanding slopes is key to identifying perpendicular lines.
  • The slopes of perpendicular lines are negative reciprocals of each other.
For example, consider a line with a slope of 2. Its perpendicular counterpart would have a slope of \(-\frac{1}{2}\). This unique slope relationship makes perpendicular lines readily identifiable. However, there's an interesting case for horizontal and vertical lines. A horizontal line has a slope of 0, while a vertical line's slope is undefined. Therefore, a horizontal line and a vertical line are perpendicular by default, due to this inherent property.
Horizontal Line
A horizontal line is one of the simplest line types in algebra and geometry. These lines run parallel to the x-axis and have an easy-to-understand equation form: \(y = b\).
  • The slope of a horizontal line is always 0.
  • The y-coordinate of every point on the line remains constant.
  • When graphed, horizontal lines appear as flat lines stretching left to right.
For an example, let's consider the line \(y = 1\). This line is horizontal because no matter what x-value you select, y is always 1. Thus, you will see a straight line parallel to the x-axis at y=1 across the entire plane.
Vertical Line
Vertical lines introduce a different perspective in linear equations. They extend infinitely in a north-south direction and have equations like \(x = a\).
  • The slope of a vertical line is undefined, which signifies that it doesn't run horizontally at all.
  • Every point on a vertical line shares the same x-coordinate but can have any y-coordinate.
  • They are represented as lines running parallel to the y-axis.
Consider a vertical line like \(x = 4\). At any point on this line, the x-value remains constant as x = 4, regardless of the y-value. This property makes it particularly unique in contrast to lines with defined slopes.
Equation of a Line
Understanding how to write the equation of a line is fundamental in algebra. Line equations often use the slope-intercept form, represented as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
  • This form is effective for lines with defined slopes.
  • For horizontal lines, since the slope \(m = 0\), the equation simplifies to \(y = b\).
  • Vertical lines are an exception since they use the form \(x = a\).
Once you determine a line鈥檚 characteristics, writing its equation becomes straightforward. For instance, if a line passes through the point (4, -1) and is vertical, the equation is simply \(x = 4\). This method enables us to express lines concisely and accurately.

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