Problem 81 - Contemporary Precalculus

Question

Do Exercise 80 with \((c, d)\) and \((c, k)\) in place of (2,1) and (2,5).

Step-by-Step Solution

Answer

Answer: The equation of the vertical line is x = c.

Step 1: Identify the given points

The given points are \((c, d)\) and \((c, k)\).

Step 2: Determine the characteristics of vertical lines

Vertical lines have the same x-coordinate for all points on the line. Therefore, the equation of a vertical line is given as \(x = c\), where \(c\) is the x-coordinate of the points on the line.

Step 3: Write the equation of the vertical line

Since the given points have the same x-coordinate \(c\), the equation of the line passing through these points is \(x = c\).

Key Concepts

Vertical Line

A vertical line is one of the simplest geometric concepts. It runs straight up and down on a graph. What's unique about a vertical line is that it has the same x-coordinate for every point along its length. Imagine a pole standing upright in the ground—no matter how high or low you go along the pole, you're still in the same spot horizontally.

This means that the equation for a vertical line is always given by a simple expression of the form \(x = c\). Here, \(c\) represents that fixed x-coordinate where the line runs.

Vertical lines are different from other types of lines, like horizontal or diagonal lines, because they don't have a slope. In fact, in algebra, vertical lines are sometimes said to have an "undefined" slope. When drawing graphs, vertical lines are very easy to identify because of their distinct up-and-down orientation.

Coordinates

Coordinates play an essential role in geometry and algebra by helping us specify the location of points on a graph. In a two-dimensional space, coordinates are given using a pair of numbers, often written as \((x, y)\).

The first number, \(x\), shows the horizontal position of the point, moving left or right from the origin which is \((0,0)\). The second number, \(y\), indicates the vertical position, moving up or down.

For vertical lines, the x-coordinate is consistent across all points on the line. For example, in our exercise, using \((c, d)\) and \((c, k)\) means that both points share the same x-coordinate \(c\). This uniformity is what defines the vertical line's equation, \(x = c\). The variation in the y-coordinates, \(d\) and \(k\), shows the line’s extent, not its direction. Coordinates are foundational in graphing, enabling precise and mathematical placement of points.

Algebra

Algebra is a branch of mathematics dealing with symbols and rules for manipulating those symbols. It's about finding the unknown and putting real-world problems into mathematical form.

One key part of algebra involves understanding equations, like those of lines. The equation of a vertical line, \(x = c\), is a basic algebraic expression. This equation tells us that no matter what the y-value is, x remains constant. In algebra, different lines and curves have unique equations that describe their shapes and positions.

Using algebra to work with geometrical concepts like lines helps us solve for unknowns, predict relationships between variables, and visualize mathematical scenarios on a coordinate plane. Algebra gives us the tools to explore various mathematical situations clearly and efficiently.

Points on a Line

Points on a line are specific locations that lie on the course of that line. When finding the equation of a line, we often start by looking at specific points, just like in the exercise where we consider two points \((c, d)\) and \((c, k)\).

For vertical lines, each of these points shares the same x-coordinate, \(c\). This is why the equation of a vertical line can be simplified to \(x = c\). These points differ only in their y-coordinate, showing how far up or down they are on that fixed vertical path.

Understanding the role of points on a line helps in graphically depicting equations and solving problems related to geometry. Points guide us in illustrating mathematical concepts and visualizing abstract ideas tangibly. This approach is crucial in both basic and complex mathematical analyses.