Problem 40 - Brief Applied Calculus

Question

Write an equation of the line satisfying the following conditions. If possible, write your answer in the form \(y=m x+b\). Vertical and passing through the point \(\left(\frac{1}{2}, \frac{3}{4}\right)\)

Step-by-Step Solution

Answer

The equation of the line is \(x = \frac{1}{2}\).

Step 1: Analyze the Problem

We need to write the equation of a line that is vertical and passes through the given point \(\left(\frac{1}{2}, \frac{3}{4}\right)\). A vertical line means it is parallel to the y-axis.

Step 2: Define the Characteristics of a Vertical Line

A vertical line in a coordinate plane has an undefined slope and is expressed as \(x = a\), where \(a\) is the constant x-value all along the line.

Step 3: Identify the x-coordinate of the Given Point

The point \(\left(\frac{1}{2}, \frac{3}{4}\right)\) has an x-coordinate of \(\frac{1}{2}\). Since the line is vertical, the x-value remains constant along the line.

Step 4: Write the Equation of the Vertical Line

Since the x-coordinate of the line is constantly \(\frac{1}{2}\), the equation of the line is \(x = \frac{1}{2}\).

Key Concepts

Vertical Line

A vertical line in geometry is a straight line that goes up and down on a graph. It is parallel to the y-axis and crosses the x-axis at a particular point. Unlike other types of lines, a vertical line does not tilt or incline, staying perfectly straight.
A key characteristic of a vertical line is that it is expressed as \(x = a\). Here, \(a\) represents a constant value, meaning every point on this line shares the same x-coordinate. For example, if a vertical line passes through the x-coordinate \(\frac{1}{2}\), the equation would be written as \(x = \frac{1}{2}\).
It's important to understand that no matter how far up or down you travel on a vertical line, the x-coordinate will never change. This is due to its parallel nature to the y-axis, making it unique on a coordinate plane.

Coordinate Plane

The coordinate plane is a fundamental element of algebra and geometry. It is a two-dimensional plane formed by the intersection of two number lines: the horizontal x-axis and the vertical y-axis.
These axes intersect at a point called the origin, which has the coordinates \((0, 0)\). Every point on this plane is represented by a pair of numbers \((x, y)\). These numbers detail how far a point is from the origin, with \(x\) indicating horizontal position and \(y\) showing vertical position.
The coordinate plane is divided into four quadrants, each representing a region with a different combination of positive and negative x and y values. Vertical lines, such as the one in our exercise, are significant because they maintain a consistent x-value across all y-values.

Undefined Slope

The concept of slope describes the steepness or incline of a line. It is calculated as the "rise" over the "run", or the change in y over the change in x. However, for vertical lines, this calculation presents a challenge.
In the case of vertical lines, the change in x (run) is zero because the line does not move left or right — it only moves up or down. When calculating the slope, if the denominator (run) is zero, we end up dividing by zero. Mathematically, division by zero is undefined, so the slope of a vertical line is said to be undefined.
This characteristic sets vertical lines apart from other lines, such as horizontal lines which have a slope of zero, and diagonal lines which have a defined slope. Understanding undefined slope is crucial in distinguishing the pattern and behavior of vertical lines on the coordinate plane.