Problem 54 - College Algebra and Calculus: An Applied Approach

Question

Find an equation of the line passing through the points. \((3,5),(3,-2)\)

Step-by-Step Solution

Answer

The equation of the line is \(x = 3\).

Step 1: Identify the common x-coordinate

Identify the common x-coordinate in both points, which is 3.

Step 2: Write the equation of the line

Since both points have the x-coordinate 3 and we established that the line is vertical, simply write the equation of the line as \(x = 3\)

Key Concepts

Linear Equations

Linear equations form the bedrock of algebra and graphing. They are mathematical statements that show two expressions set equal to each other, typically in the form of \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

However, not all linear equations fit neatly into this format. When dealing with vertical lines, the equation takes a unique form because vertical lines have an undefined slope. As such, the equation of a vertical line is simply \( x = a \), where \( a \) is the constant x-coordinate for all points on the line. This is because a vertical line does not change in the y-direction – no matter how high or low you go on the line, the x-coordinate remains the same.

The vertical line equation differs from the standard linear equation due to the slope being infinite; thus, the equation eliminates the y variable and slope altogether, simplifying to a relation reflecting a constant x-value.

Coordinates in the Cartesian Plane

The Cartesian plane is a two-dimensional grid formed by the intersection of a vertical line (y-axis) and a horizontal line (x-axis). This intersection is known as the origin, where both x and y coordinates are zero. The position of any point on this plane is determined by its x and y coordinates, represented as an ordered pair \( (x, y) \).

A crucial aspect of understanding the Cartesian plane is recognizing that every point has a specific location based on these two values. The x-coordinate reflects horizontal movement from the origin, while the y-coordinate represents vertical movement. The coordinate system is divided into four quadrants, each representing a combination of positive and negative values for x and y.

For a vertical line, all points share the same x-coordinate. For instance, in our exercise, the points \( (3, 5) \) and \( (3, -2) \) lie on the same vertical line because their x-coordinate is 3. This reflects how, on the Cartesian plane, a vertical line moves along the y-axis at a constant x value.

Properties of Vertical Lines

Vertical lines have unique properties that differentiate them from other types of lines on the Cartesian plane. One of the key properties to note is that vertical lines have undefined slopes. This means that their incline or descent cannot be measured because they rise infinitely without running horizontally.

This undefined slope leads to the specific form of the equation for a vertical line, which, as established, is expressed as \( x = k \), where \( k \) is the constant x-coordinate of any point on the line. Therefore, vertical lines don't form a function in the traditional sense because they do not pass the vertical line test; they contain more than one y-value for a singular x-value.

Furthermore, no two different vertical lines intersect, which means they are parallel to each other. This property makes the concept of vertical lines a powerful tool in geometry and helps in understanding the structure of space within the Cartesian coordinate system.