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Chapter 12: Problem 64

What is the form of the equation of a horizontal line? Of a vertical line?

Short Answer

Expert verified
Horizontal line: \(y = c\); Vertical line: \(x = c\).

Step by step solution

01

Understanding Horizontal Lines

A horizontal line is a straight line that runs from left to right across the graph. A key characteristic of a horizontal line is that it has a constant y-coordinate, regardless of the value of the x-coordinate. This means that the y-value doesn't change as you move along the line.
02

Equation of a Horizontal Line

Since the y-coordinate remains constant for a horizontal line, the equation is of the form \(y = c\), where \(c\) is the constant y-value of every point on the line.
03

Understanding Vertical Lines

A vertical line is a straight line that runs up and down the graph. A key characteristic of a vertical line is that it has a constant x-coordinate, regardless of the value of the y-coordinate. This means that the x-value doesn't change as you move along the line.
04

Equation of a Vertical Line

Since the x-coordinate remains constant for a vertical line, the equation is of the form \(x = c\), where \(c\) is the constant x-value of every point on the line.

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Key Concepts

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

Understanding Horizontal Lines
In coordinate geometry, a horizontal line is quite simple to visualize. It's like imagining a ladder lying flat on the ground. It extends infinitely to the left and right but stays at the same height.

A key thing about horizontal lines is their constant y-coordinate. For example, if we say "the line is at y = 3", the line will cut through all points where the y-value is 3, such as (1, 3), (2, 3), (3, 3), and so on. This means:
  • The y-coordinate is the same for every point.
  • The line doesn't move up or down, only side to side.
  • Represented by the equation: \(y = c\)
Understanding this will help you quickly identify and write the equation for any horizontal line you encounter.
Understanding Vertical Lines
Vertical lines run from the ground to the sky as if you're staring at a tall building. They extend upwards and downwards infinitely, keeping their position tightly fixed along the x-axis.

The constant feature of a vertical line is its steadfast x-coordinate. For example, if we say "the line is at x = 4", this indicates the line will pass through points like (4, 1), (4, 2), (4, 3), and so forth. This implies:
  • The x-coordinate doesn't change regardless of the y-value.
  • The line only moves up and down, not side to side.
  • Described by the equation: \(x = c\)
Knowing this makes it straightforward to write the equation for vertical lines, as you'll always use the x-value that stays constant.
Coordinate Geometry in Context
Coordinate geometry is a method of representing geometrical shapes using algebra in a two-dimensional plane. A Cartesian coordinate system helps us plot and analyze these shapes efficiently. Here鈥檚 a look at how it relates to lines:

In the case of straight lines, coordinate geometry allows us to precisely describe the placement and direction of a line utilizing equations. With the Cartesian system, each point is defined by an x-coordinate and a y-coordinate, which determines the position on the graph.
  • Horizontal and vertical lines are specific examples that show constant behaviors along one axis.
  • Understanding these lines helps in solving more complex geometrical problems.
As you delve deeper into coordinate geometry, recognizing these line types becomes crucial in mastering topics like graph transformations, intersections, and slopes.

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Most popular questions from this chapter

Use Cramer's rule to solve system of equations. If a system is inconsistent or if the equations are dependent, so indicate. \(\left\\{\begin{array}{l}2 x+3 y=0 \\ 4 x-6 y=-4\end{array}\right.\)

When solving a system, what advantages are there with the substitution and elimination methods compared with the graphing method?

Which method would you use to solve the system? Explain. $$ \left\\{\begin{array}{l} y-1=3 x \\ 3 x+2 y=12 \end{array}\right. $$

Solve each formula for the specified variable. $$ \frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}} \text { for } r $$

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) $$ \left\\{\begin{array}{l} x=2 \\ y=-\frac{1}{2} x+2 \end{array}\right. $$
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