Problem 17 - Calculus for Biology and Medicine

Question

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The vertical line through \((-2,0)\)

Step-by-Step Solution

Answer

The equation is x = -2.

Step 1: Identify Characteristics of a Vertical Line

A vertical line has an undefined slope and does not depend on the y-coordinate. It is represented by an equation where x equals a constant.

Step 2: Use Point Coordinates to Define the Line

The given point through which the vertical line passes is (-2,0). This means that for any point on this line, the x-coordinate will always be -2.

Step 3: Write the Equation of the Line in Standard Form

The equation of a vertical line through the point (-2,0) is x = -2. In standard form, a line's equation is expressed as Ax + By = C. For a vertical line, only the x-variable is present, so the equation in standard form is x - 0y = -2, which simplifies to x = -2.

Key Concepts

Standard Form

When we discuss the standard form of a line's equation, we're referring to a specific way to express linear equations. In this format, equations are rearranged into:

• Ax + By = C

where A, B, and C are integers, and A is usually non-negative. This form is particularly useful for quickly identifying intercepts and for computational applications.
For a vertical line, the term involving the y-variable is absent. So, the equation is simplified as Ax = C because a vertical line doesn't change with varying y-values. A typical vertical line like x = -2 would be written in standard form as x + 0y = -2. It highlights that y's coefficient is zero, confirming the line's vertical nature.

Equation of a Line

The equation of a line is a mathematical representation that describes all the points lying on that line. Lines can be categorized based on their direction and are usually represented in various forms, including:

• Point-slope form
• Slope-intercept form
• Standard form

Vertical lines are unique because they don't fit into the more common slope-intercept form (y = mx + b) since their slope ( ) is undefined. Thus, their equation is represented as a simple equality involving the x-coordinate.
For instance, the line through (-2,0) is merely x = -2. All points on this line share the x-coordinate value of -2, regardless of their y-values. This simplicity makes vertical lines straightforward to work with once recognized.

Undefined Slope

The concept of a slope gives insight into a line’s steepness and direction. Calculating the slope generally involves determining the ratio of change in y-coordinates to the change in x-coordinates between two points. This is often written as:

• slope = \( \frac{\Delta y}{\Delta x} \)

However, for vertical lines, there is no horizontal change (\Delta x), meaning we attempt to divide by zero. This division is mathematically undefined, which is why vertical lines have an undefined slope.
Despite having an undefined slope, vertical lines can still be described algebraically with their x-coordinate constant for all points, leading to easy identification and understanding. As you explore linear equations, recognizing when a slope is undefined will help in classifying and writing special types of line equations, particularly those describing vertical lines.