Problem 39 - Calculus for Biology and Medicine

Question

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((1,-3)\) and perpendicular to the horizontal line passing through \((-1,-1)\)

Step-by-Step Solution

Answer

The equation of the line in standard form is \(x = 1\).

Step 1: Understanding a Horizontal Line

A horizontal line has a slope of 0. A line perpendicular to a horizontal line will have an undefined slope, meaning it is a vertical line.

Step 2: Vertical Line Equation

Since the line is vertical and passes through the point \((1, -3)\), its equation is in the form \(x = a\). For this line, \(a = 1\). Thus the equation of the line is \(x = 1\).

Step 3: Converting to Standard Form

The standard form of a line is \(Ax + By = C\). For a vertical line, the equation can be written as \(x = 1 \rightarrow 1x + 0y = 1\), which is \(x = 1\).

Key Concepts

Vertical Lines

Vertical lines are special types of lines on the Cartesian plane that run up and down. They possess a unique characteristic: they do not have a defined slope. This is because slope is calculated as "rise over run," and for a vertical line, the "run" or the horizontal change is zero. Attempting to divide by zero makes the slope undefined.
Vertical lines can be represented by an equation of the form \(x = a\), where \(a\) is the x-coordinate through which the line passes. This means every point on a vertical line shares the same x-coordinate. For example, if the line passes through the point \((1, -3)\), then the vertical line can be expressed as \(x = 1\).
Understanding vertical lines is crucial when discussing their perpendicular relationship to horizontal lines, as they help in identifying orientation and slopes on a graph.

Standard Form Equation

The standard form equation of a line is represented by \(Ax + By = C\). Here, \(A\), \(B\), and \(C\) are integers, and \(A\) should be a non-negative integer. This form is quite versatile and often used for its clarity, especially when analyzing and graphing linear equations.
When it comes to vertical lines, the standard form takes on a slightly different appearance. Since vertical lines have an undefined slope, the equation \(x = a\) can be converted into standard form as \(1x + 0y = a\). This keeps \(A\) as 1, \(B\) as 0, and \(C\) as the same x-value where the line crosses, maintaining the integrity of the vertical line equation. Thus, for a vertical line crossing \(x = 1\), the standard form becomes \(x = 1\).
Using this form can simplify solving systems of equations and is a great tool for graphing purposes.

Slope

Slope is a measure of how steep a line is on a graph. It is calculated as the change in the y-coordinates divided by the change in the x-coordinates between two points on a line. The formula for slope is \(m = \frac{\Delta y}{\Delta x}\), where \(\Delta y\) is the difference in y-values and \(\Delta x\) is the difference in x-values.
For most lines, this provides a reliable measure of incline, but for vertical lines, the slope is undefined. Imagine a graph with points stacked directly above each other on the x-axis. Since there's no horizontal movement between these points, \(\Delta x = 0\), making \(\frac{\Delta y}{0}\) impossible to calculate.
In contrast, the slope of a horizontal line is always 0, since there is no vertical change (\(\Delta y = 0\)). Understanding these extremes helps in identifying perpendicular lines on a graph, as a vertical line will always be perpendicular to a horizontal line.