Problem 16 - Calculus for Biology and Medicine

Question

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The horizontal line through \((0,-1)\)

Step-by-Step Solution

Answer

The equation of the horizontal line is 0x + 1y = -1.

Step 1: Understand the Definition of a Horizontal Line

A horizontal line is one that runs left to right and has a constant y-coordinate value for all points on the line.

Step 2: Identify the Given Point

We are given the point (0, -1) which means it is on the line and has a y-coordinate of -1.

Step 3: Determine the Equation of the Line

Since a horizontal line keeps the y-coordinate constant, the equation of our line will simply be y = -1.

Step 4: Convert to Standard Form

The standard form of a line is Ax + By = C. For a horizontal line, y = -1 can be rewritten in standard form as 0x + 1y = -1.

Key Concepts

Standard Form of a Line

When we talk about lines in mathematics, especially in coordinate geometry, we often need to represent them in a specific format called the standard form. This form is useful because it provides a uniform way to describe lines, making them easier to compare and analyze. The standard form of a line is represented as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers.

• \( A \), \( B \), and \( C \) should have no common factors other than 1, ensuring the equation is simplified.
• \( A \) and \( B \) can’t both be zero; otherwise, it wouldn’t describe a line.

To convert the equation of a line to its standard form, algebraic manipulation is typically required. This helps in creating an equation where all terms are neatly organized. For instance, for a horizontal line like \( y = -1 \), we manipulate it to look like \( 0x + 1y = -1 \) to fit the \( Ax + By = C \) format.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, bridges algebra and geometry through graphs of lines and curves. This branch of geometry involves using coordinates, usually \( (x,y) \), to represent geometrical shapes on a plane.

• It helps us understand spatial relationships and geometric properties in a visual way.
• Using coordinates, we can find the distance between points, the slope of a line, and the midpoint of a segment.

In the context of the exercise, we use coordinate geometry to understand the positioning of the horizontal line and the point \( (0, -1) \) through which it passes. By knowing this, we can determine that the line's equation is simply \( y = -1 \), as the y-coordinate is constant, illustrating one of the key concepts of coordinate geometry.

Equation of a Line

The equation of a line is a mathematical expression that describes all the points lying on that line. Each point on the line satisfies the equation.A line can be described using different forms, such as the slope-intercept form, point-slope form, and standard form. For a horizontal line, like our example \( y = -1 \), it doesn't depend on the x-coordinate, which simplifies the equation significantly.

• In general, a line's equation can take the form \( y = mx + c \) in slope-intercept form, where \( m \) is the slope, and \( c \) is the y-intercept.
• A horizontal line simplifies this to \( y = c \), where \( c \) is the constant y-coordinate throughout.

Thus, the equation \( y = -1 \) shows that for every point on this line, the y-coordinate remains \(-1\). Transforming this into the standard form involves expressing it as \( 0x + 1y = -1 \), clearly showcasing that the equation applied is equal for all points on the line.