Problem 55 - College Algebra with Modeling and Visualization

Question

Find an equation of the line satisfying the conditions. Parallel to \(x=4.5,\) passing through \((19,5.5)\)

Step-by-Step Solution

Answer

The equation is \(x = 19\).

Step 1: Understanding the Problem

We need to find an equation of a line that is parallel to the vertical line \(x = 4.5\) and passes through the point \((19, 5.5)\). Vertical lines are of the form \(x = c\), where \(c\) is a constant.

Step 2: Identify the Type of Line

Since we need a line parallel to \(x = 4.5\), our line must also be vertical. A vertical line is characterized by having all points with the same x-coordinate.

Step 3: Using the Point to Define the Line

The line must pass through the point \((19, 5.5)\). For a vertical line, this means the x-coordinate of every point on the line is 19.

Step 4: Write the Equation

Since the line is vertical and all points on it share the x-coordinate of 19, the equation of the line is \(x = 19\).

Key Concepts

Equations of Lines

In algebra, understanding lines and their equations is fundamental. An equation of a line describes all the points that lie on that line in a coordinate plane. A basic form to express this is the slope-intercept form, which is written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. However, not all lines can be written in this form. For instance, vertical lines don’t have a defined slope, and hence, their equations are a bit different.

When considering the slope-intercept form, the concept of slope refers to how steep the line is. For horizontal lines, the slope is zero, which makes the equation of the line in the form \(y = b\), where \(b\) is constant for all points on the line. But when dealing with vertical lines, things change significantly since the slope is undefined.

Understanding these variations is crucial because lines can be oriented in various directions, and the representation of their equations must accommodate these differences.

Vertical Lines

Vertical lines are unique in the world of algebra because they don't have a slope like other lines do. Instead, their equations are written in the form \(x = c\), where \(c\) is a constant number representing the x-coordinate of all points on the line. This is because, for vertical lines, every point along the line shares the same x-coordinate.

For instance, if you're given the line \(x = 4.5\), this tells you that no matter the y-coordinate, the x-coordinate will always be 4.5, creating a perfectly vertical line on a graph.

This idea of a constant x-coordinate is the defining characteristic of a vertical line. This differs from the more commonly visualized horizontal lines, which maintain a constant y-coordinate. Vertical lines are parallel to the y-axis and are critical when determining the position of a point relative to vertical boundaries.

Parallel Lines

Parallel lines are lines in a plane that never intersect. They remain the same distance apart over their entire length, which means their slopes are equal. This property is crucial in problems that involve finding one line parallel to another.

In the context of vertical lines, any line parallel to another vertical line is also vertical. Since vertical lines don’t have defined slopes, they are more simply described by their constant x-values. For example, a line parallel to \(x = 4.5\) will have its equation as \(x = c\), where \(c\) is different from 4.5, specific to the conditions set by the problem.

In summary, understanding parallel lines is essential, particularly in determining when two lines will never meet. This is a fundamental concept in geometric studies and plays a significant role in more complex algebraic problems.