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 standard form of the equation is \(0x + 1y = -1\).

Step 1: Understand the Characteristics of a Horizontal Line

A horizontal line is a line where all the points have the same y-coordinate. Therefore, no matter what the x-coordinate is, the y-value remains constant.

Step 2: Identify the Constant Y-Value

Since the line goes through the point \((0, -1)\), it means that the y-coordinate for all points on this line is \(-1\). So, the equation of the horizontal line is simply \(y = -1\).

Step 3: Convert to Standard Form

The standard form of a linear equation is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) should be non-negative. Since \(y = -1\) is a horizontal line, it can be rewritten in standard form as \(0x + 1y = -1\), or simplified as \(y = -1\).

Key Concepts

Horizontal Line

A horizontal line is a type of line where all points lie in a straight path parallel to the x-axis. Thus, each point on a horizontal line shares the same y-coordinate, meaning the y-value remains constant regardless of any variation in the x-coordinate.

For example, if a line is horizontal and passes through the point \((0, -1)\), its y-coordinate remains \(-1\) for any x-value.
This property makes horizontal lines special, as they essentially have a slope of zero. Because the slope, calculated as the change in y over the change in x, shows no change in y for horizontal lines, the result is zero slope.

To determine the equation of a horizontal line, simply identify this constant y-value. Therefore, the equation for our horizontal line is simple: \(y = -1\). This equation shows that no matter the x-value, the y will always be \(-1\).

Standard Form of a Linear Equation

The standard form of a linear equation is a specific way of writing the equation of a line. It helps provide a uniform method to handle different line types through an equation of the form \Ax + By = C\. Here, \A\, \B\, and \C\ are integers, with the condition that \A\ should not be negative unless zero.

In our case, we started with \(y = -1\), a typical representation of a horizontal line. To convert this into standard form, check that both x and y are represented. However, horizontal lines do not depend on x, so their equations do not include an x-term.
The horizontal line through \(0, -1)\) simplifies into standard form by introducing an implicit zero-multiplied x-term: \(0x + 1y = -1\).

Therefore, the line given by \(y = -1\) naturally fits the standard form of \(Ax + By = C\), often without further simplification, as \(y = -1\).

Linear Equations in Two Variables

Linear equations in two variables are foundational in representing lines on a graph. They express the relationship between two variables, typically represented as \x\ and \y\, and form a straight line when plotted.

Such equations usually follow the form \Ax + By = C\ or the slope-intercept form \y = mx + b\.
• **Form Elements:** Where \x\ and \y\ are variables, \A\ and \B\ are coefficients representing the line's direction, and \C\ is the constant term, defining its location on the graph.
• **Graphing:** Every solution pair (x, y) to this equation corresponds to a point on the line.
• **Properties:** This includes its slope (rise over run) and intercepts with the axes.

In simpler terms, a linear equation in two variables depicts how y changes in response to x. If we return to our horizontal example \(0x + 1y = -1\), the lack of x's influence simplifies our model.
The equation reduces to a horizontal line, showing that vertical change is nil and x's role is limited. This emphasizes how the relationship between x and y dictates the line's orientation in the coordinate plane.