Problem 47 - Algebra A Combined Function

Question

Write an equation of each line. Write the equation in standard form unless indicated otherwise. See Examples 1 through \(6 .\) Vertical line; through (-2,-10)

Step-by-Step Solution

Answer

The equation of the vertical line is \(x = -2\).

Step 1: Identify the Characteristics of the Line

A vertical line has a constant x-coordinate throughout its length. This means that no matter the y-coordinate, the x-coordinate remains the same.

Step 2: Identify the Given Point

The given point through which the vertical line passes is (-2,-10). For a vertical line, the x-coordinate is constant, which is -2 in this case.

Step 3: Write the Equation of the Vertical Line

Since the x-coordinate is constant for any vertical line, the equation of the line can be written as \(x = -2\).

Step 4: Convert to Standard Form (If Possible)

The standard form for a line is \(Ax + By = C\). In the case of a vertical line like \(x = -2\), it is already in a form closely resembling standard form: \(1x + 0y = -2\).

Key Concepts

Standard Form of a Line

When discussing lines in algebra, each line can often be represented in what is known as "standard form." This form utilizes a specific equation format: \[ Ax + By = C \]where:

• \( A \), \( B \), and \( C \) are integers.
• \( A \) and \( B \) cannot both be zero.
• The coefficients are typically simplified so that \( A \) is a non-negative number.

This standard format is beneficial for easily identifying the various elements and characteristics of a line, including the slope and intercepts.

In the context of vertical lines, the standard form can seem a bit different, since vertical lines do not have a slope. A vertical line equation like \( x = -2 \) can be rewritten in a format that fits the standard form: \[ 1x + 0y = -2 \]This illustrates that while the form seems less intuitive, it allows vertical lines to fit within this broader framework of line equations.

x-coordinate

The x-coordinate is a critical part of any point on a Cartesian plane. It signifies the horizontal position of a point relative to the origin. In the pair \((-2, -10)\), \(-2\) is the x-coordinate.

For vertical lines, the x-coordinate is especially important because it remains constant along the line. Regardless of the y-coordinate, every point on a vertical line through \((-2, -10)\) (or any point where the x-coordinate is \(-2\)) shares that same x-coordinate: \(-2\).This consistency makes vertical lines unique and simple to describe.

Understanding how the x-coordinate operates is crucial when writing the equation for a vertical line. You'll note that unlike other lines, vertical lines are not defined by y-coordinates, but solely by this unchanging x-coordinate value.

Line Equation

A line equation allows us to express the set of all points that lie on a particular line. For most lines, an equation can be expressed in forms like slope-intercept or point-slope. However, vertical lines, like the one through \((-2, -10)\),have their own unique characteristics.

Vertical lines have x-coordinates that do not change. Therefore, their equations are straightforward. A vertical line that passes through any specified x-coordinate can be written as:\[ x = a \]where \( a \) is the constant x-coordinate for all points on the line.
In this case, the line through the point \((-2, -10)\) can be expressed by the equation:\[ x = -2 \]This equation highlights that, unlike other kinds of lines, vertical lines are defined purely by their x-coordinate, reflecting their constant nature.