Problem 30 - College Algebra

Question

For the following exercises, find the equation of the line using the given information. \((-2,0)\) and \((-2,5)\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Type of Line

Given that both points have the same x-coordinate, (-2,0) and (-2,5), this indicates that the line is vertical.

Step 2: Confirm Vertical Line Equation

For vertical lines, the equation is of the form \(x = a\), where \(a\) is the x-coordinate of the points. Since both given points have \(x = -2\), the equation of the line is \(x = -2\).

Key Concepts

Vertical Line Equation

A vertical line is a straight line that moves only up and down without changing its horizontal position. In coordinate geometry, vertical lines have a unique and simple equation: \(x = a\). Here, \(a\) represents the constant x-coordinate for all points on the line.

Vertical lines differ from horizontal or diagonal lines because they have an undefined slope. This is due to the fact that the change in the y-values occurs while the x-value remains constant. When describing a vertical line, the equation \(x = a\) means that for every point along this line, the x-coordinate is always \(a\).

In the provided exercise, both points (-2,0) and (-2,5) lie on the line with a constant x-coordinate of \(-2\). Thus, the vertical line equation is \(x = -2\).

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses numbers to describe spatial relationships. This approach allows us to use algebraic formulas and equations to solve geometric problems.

In coordinate geometry, we often plot points on a plane using pairs of numbers known as coordinates. The first number represents the x-coordinate (horizontal position), and the second number represents the y-coordinate (vertical position). Utilizing these coordinates, lines and shapes can be graphically represented and their properties can be derived.

For example, the line through points (-2,0) and (-2,5) showcases how coordinate geometry is used to find the equation of a line. By locating these points on the graph and considering their properties, we determine that they form a vertical line with the equation \(x = -2\). This is a practical application of coordinate geometry principles.

X-Coordinate

The x-coordinate is a fundamental element of a point in a two-dimensional Cartesian coordinate system. It denotes the horizontal position of a point relative to the origin (0,0).

In any coordinate pair, the x-coordinate comes first and is followed by the y-coordinate. For example, in the coordinate pairs (-2,0) and (-2,5), the x-coordinate is \(-2\). The x-coordinate informs us how far left or right a point is from the vertical y-axis.

The role of the x-coordinate in identifying vertical lines is crucial. When two points share the same x-coordinate, as in this exercise, it immediately indicates a vertical line. Such lines will always have an equation of the form \(x = a\), where \(a\) is the common x-coordinate. Hence, points like (-2,0) and (-2,5) residing on the line with x-coordinate \(-2\) lead us to the equation \(x = -2\). This simplicity is one of the appealing aspects of mastering the concept of x-coordinates.