Problem 18 - Calculus for Biology and Medicine

Question

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The vertical line through \((2,-3)\)

Step-by-Step Solution

Answer

The equation of the line in standard form is \(x - 2 = 0\).

Step 1: Understanding Vertical Lines

A vertical line is a line where all points on the line have the same x-coordinate. This means that no matter what the y-coordinate is, the x-coordinate remains constant.

Step 2: Identify the Given x-coordinate

The problem states the line goes through the point \((2, -3)\). Since it is a vertical line, this x-coordinate (2) will apply for all points on the line.

Step 3: Write the Equation

For a vertical line with constant x-coordinate, the equation is simply: \[ x = a \]where \(a\) is the x-coordinate. Here, it is \(x = 2\).

Step 4: Conversion to Standard Form

The standard form of a line is usually written as \[Ax + By = C\]. For vertical lines, the standard form is slightly different because \(B = 0\). Hence, the line \(x = 2\) can be expressed in standard form as:\[x - 2 = 0\].

Key Concepts

Standard Form of Line

Lines can be expressed in different forms, allowing for various interpretations and uses, particularly in algebra and geometry. One such form is the "standard form," which is useful for capturing the essence of both horizontal and vertical line equations.
In general, the standard form of a line is given by: \ \[ Ax + By = C \] where:

• \(A\), \(B\), and \(C\) are integers.
• \(x\) and \(y\) are the variables representing coordinates.

For vertical lines, however, the standard form is specially adjusted because these lines have an undefined slope. This means that the change in \(y\) over the change in \(x\) is infinite, leading to a situation where \(B = 0\) in the original equation.
Therefore, the simplified standard form for a vertical line, like \(x = 2\), becomes \(x - 2 = 0\). This keeps "\(x\)" isolated, showing its constancy along this type of line.

Coordinates

Coordinates are essential in locating points in a plane. They serve as a pair of numbers used to define the position of points, often in a two-dimensional plane. In coordinate geometry, we use ordered pairs:

• The first number, known as the "x-coordinate," indicates the position along the horizontal axis.
• The second number, known as the "y-coordinate," specifies the location on the vertical axis.

In the context of vertical lines, like the problem statement, coordinates gain unique significance. An example given is the point \((2, -3)\). Here, the x-coordinate is consistent across all points on a vertical line, while the y-coordinate can vary. This regularity in the x-coordinate is what defines the line as vertical, with the equation \(x = 2\).

Geometry

Geometry, as a branch of mathematics, deals with size, shape, relative position, and properties of space. It plays a significant role in understanding concepts related to lines, shapes, and angles. In our context, the geometry of lines, especially vertical lines, is of interest.
A vertical line is notable in geometry for several reasons:

• Vertical lines run perpendicular to the horizontal axis, extending in the vertical direction.
• They are associated with a fixed x-coordinate across all points on the line, rendering them parallel to the y-axis.
• They illustrate the principle that for vertical lines, the slope (rate of change of y with respect to x) is undefined, reinforcing the irregularity seen in their standard form equation.

Understanding these properties help in grasping why a line like \(x = 2\) maintains its vertical nature, making it distinct from lines with varying y-coordinates.