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Chapter 4: Problem 491

Find the equation of each line. Write the equation in slope-intercept form. Perpendicular to the line \(y\) -axis, point (-6,2)

Short Answer

Expert verified
y = 2

Step by step solution

01

Identify the Characteristics of the Given Line

To find the equation of a line perpendicular to the y-axis, observe that any line perpendicular to the y-axis is a horizontal line.
02

Recognize the Horizontal Line Property

A horizontal line has a constant y-coordinate for all x-values throughout the line. Therefore, the equation of the line will be in the form of y = k, where k is a constant.
03

Determine the y-coordinate of the Given Point

Since the line must pass through the point (-6,2), the y-coordinate of the point is 2. This means the constant k in the line equation is 2.
04

Write the Equation in Slope-Intercept Form

With k identified as 2, the equation of the line is y = 2.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a line's equation is a fundamental concept in algebra. It is written as:
\( y = mx + b \).
  • Here, m stands for the slope of the line, which represents the rise over run or the steepness of the line.
  • The term b is the y-intercept, representing the point at which the line crosses the y-axis.
Using this form makes it easy to quickly understand the line's behavior and its intersections with the axes. For a horizontal line, the slope m is zero, simplifying the equation to y = b, where b is a constant and represents the y-coordinate of any point on the line.
Horizontal Lines
Horizontal lines have some distinct characteristics. Firstly, their equation is simple because their slope is zero:
\( y = b \)
  • In the equation, y remains constant, meaning the line extends horizontally, with all points on the line sharing the same y-coordinate.
  • This feature leads to the line being flat and parallel to the x-axis.
  • An important thing to remember is that any horizontal line is perpendicular to the y-axis.
For instance, if we have a point (-6, 2), the horizontal line passing through this point has the equation y = 2, as the y-coordinate of the point is 2.
Perpendicular Lines
Perpendicular lines intersect at right angles (90 degrees). Their slopes are related in a special way:
  • The product of their slopes is -1. For instance, if one line has a slope of m, the line perpendicular to it will have a slope of \( -\frac{1}{m} \).
  • A horizontal line, as mentioned, has a slope of 0. Because any number multiplied by 0 is 0, lines perpendicular to horizontal lines are vertical lines.
For this exercise, since we need a line perpendicular to the y-axis, we refer to horizontal lines, as they intersect the y-axis at a 90-degree angle. So, for a point like (-6, 2), the perpendicular horizontal line will have the equation y = 2.

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