Problem 40 - Intermediate Algebra

Question

Contains the point \((-4,7)\) and is perpendicular to the \(x\) axis

Step-by-Step Solution

Answer

The equation of the line is x = -4.

Step 1: Identify the Key Condition

The line needs to be perpendicular to the x-axis. A line perpendicular to the x-axis is a vertical line.

Step 2: Understand the Characteristics of Vertical Lines

Vertical lines have a constant x-coordinate, which means that the x-value is the same for all points on the line.

Step 3: Determine the Equation of the Line

Since the line is vertical and contains the point (-4,7), it must have the x-coordinate of -4 for all its points. Therefore, the equation of the line is x = -4.

Key Concepts

Vertical Lines

Vertical lines are a special type of line in coordinate geometry. They have a unique characteristic where all points along the line share the same x-coordinate. This means the line does not have a slope like diagonal lines do. Since there is no change in the x-values, vertical lines run parallel to the y-axis. In simpler terms, imagine drawing a straight line that goes up and down without tilting to the sides.
Here are some key points about vertical lines:

• A vertical line has an undefined slope. This is because slope is calculated as the rise over the run (change in y over change in x), and in a vertical line, the change in x is zero, which makes the slope undefined.
• The equation of a vertical line can always be represented as \(x = a\), where "a" is the constant x-coordinate for every point on the line.
• Vertical lines are always perpendicular to horizontal lines, which run parallel to the x-axis.

Equation of a Line

An equation of a line is a mathematical way to represent all the points along a line on a coordinate plane. It provides a rule that every point on the line follows. The most common form is the slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
However, vertical lines do not fit into this form because their slopes are undefined. For a vertical line, the equation is different:

• For vertical lines, the equation is simply \(x = a\), where "a" is the constant value of the x-coordinate for the line. In this case, if a line passes through the point \((-4, 7)\), the equation is \(x = -4\).

This means that no matter where you move along this line, the x-coordinate remains the same, reinforcing the concept of verticality. It doesn't intersect the y-axis, and its definition is easy and straightforward.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra with geometry using a coordinate system. It allows us to express geometric properties through algebraic equations and vice versa.
Coordinate geometry involves:

• Points, which are represented by pairs of numbers (x, y), showing their position on the Cartesian plane.
• Lines, which are defined by equations showing the relation between x and y coordinates of any point lying on the line.
• Shapes, which can be graphically represented and analyzed using equations, often revealing deeper properties like sizes and angles.

A key strength of coordinate geometry is its ability to help us understand relationships between geometric figures and their equations. For instance, it clarifies when two lines are parallel, perpendicular, or intersecting. The simplicity and power of coordinate geometry allow us to solve problems involving distances, midpoints, and slopes, and understand the nature of vertical and horizontal lines effortlessly.