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Chapter 2: Problem 78

Find an equation of each line. Vertical line through \((14, 122)\)

Short Answer

Expert verified
The equation of the vertical line is x = 14.

Step by step solution

01

Understand the Characteristics of Vertical Lines

Vertical lines run parallel to the y-axis and have an undefined slope. The equation of a vertical line can be represented as x = a, where a is the x-coordinate of all points on the line.
02

Identify the Given Point

The given point through which the vertical line passes is (14, 122). In this case, the x-coordinate is 14.
03

Write the Equation of the Vertical Line

Since vertical lines have the same x-coordinate for all points on the line, the equation is simply x = 14.

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Key Concepts

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

Vertical Lines
Vertical lines are a fundamental concept in algebra and coordinate geometry. They run parallel to the y-axis and stand straight up and down on a graph. These lines are unique because, for any given vertical line, the x-coordinate of all points on the line remains constant, while the y-coordinates can vary. This means that no matter how high or low you go along the line, the x-value does not change. Understanding vertical lines is crucial because they help in plotting coordinates accurately and they behave differently compared to other types of lines, such as horizontal or sloped lines.
Equation of a Line
The equation of a line is a mathematical expression that shows the relationship between the x and y coordinates of points on the line. For most lines, the equation can be represented in the slope-intercept form, which is y = mx + c. However, for vertical lines, this form does not work. Instead, vertical lines have a unique equation format: x = a, where a is the constant x-coordinate of every point on the line. This equation shows that no matter what the y-value is, the x-value will always stay the same.
Undefined Slope
In coordinate geometry, the slope of a line measures its steepness and direction. Most lines have a defined slope, calculated using the formula m = (y2 - y1) / (x2 - x1). However, vertical lines are an exception. Because the x-coordinates are identical for all points on a vertical line, the denominator in the slope formula is zero, making the slope undefined. An undefined slope indicates a vertical line that does not tilt left or right but goes straight up or down. This unique characteristic of vertical lines makes them easy to identify in equations and graphs.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is the study of geometric figures using a coordinate plane. It combines algebra and geometry to understand the properties and relationships of figures through equations. In coordinate geometry:
  • Points are defined by coordinates (x, y).
  • Lines and curves are described with equations.
  • Geometric shapes and their properties are analyzed using algebraic methods.
Understanding vertical lines within this context is important because it shows how algebra can describe geometric concepts. For example, a vertical line through the point (14, 122) in a coordinate plane is simply written as x = 14. This equation helps visualize and solve problems involving vertical lines quickly and efficiently.

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Most popular questions from this chapter

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