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 \(\left(-1, \frac{7}{2}\right)\)

Step-by-Step Solution

Answer

The equation is \(x = -1\).

Step 1: Understand the Problem

We need to find the equation of a vertical line that passes through the point \((-1, \frac{7}{2})\). A vertical line means it is parallel to the y-axis and has the same x-coordinate value for every y-value along the line.

Step 2: Identify the Key Characteristic of a Vertical Line

The key property of a vertical line is that its equation is in the form \(x = a\), where \(a\) is a constant. In this problem, \(a\) is the x-coordinate of the given point.

Step 3: Write the Equation of the Vertical Line

Since the point provided is \((-1, \frac{7}{2})\), the x-coordinate is \(-1\). Therefore, the equation of the vertical line is \(x = -1\).

Step 4: Convert the Equation to Standard Form

In standard form, a linear equation is written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) is non-negative. For a vertical line like \(x = -1\), it can be rewritten in standard form as \(x + 0\cdot y = -1\) or simply \(x = -1\).

Key Concepts

Vertical Line

A vertical line is a special type of line in geometry and algebra, characterized by having a constant x-coordinate despite varying y-coordinates. This happens because the line runs parallel to the y-axis. Therefore, all the points on a vertical line have the same x-coordinate but different y-coordinates. A simple way to visualize a vertical line is to imagine line markers on a graph standing straight up and down.

The equation of a vertical line can be summarized with the formula:

• \( x = a \), where \( a \) is the x-coordinate that all points on the line share.

This is in contrast to horizontal lines where the y-coordinate is constant, and the general formula is \( y = b \). Vertical lines cannot be expressed in the slope-intercept form (\( y = mx + b \)) because their slope is undefined or infinite.

Vertical lines are often seen on graphs when a boundary or fixed x-value needs to be represented. These lines are simple yet crucial in understanding the different orientations lines can take in a coordinate plane.

Standard Form

In algebra, the standard form of a linear equation is provided as:

• \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integer constants,\( x \) and \( y \) are the variables.
• \( A \) should preferably be a non-negative integer.

Standard form is quite versatile because it not only helps in writing equations of lines but also facilitates easy identification of the x-intercept and y-intercept. When it's applied to vertical lines, however, the variable \( y \) drops out.

For a vertical line, like \( x = -1 \), converting to standard form is straightforward. Here, no \( y \)-term is needed, hence its equation can remain as \( x + 0 \cdot y = -1 \) or simply \( x = -1 \). This matches the general structure of standard form where \( By \) can simply be \(0\) since there isn't any change in the y-variable across the line.

Linear Equation

Linear equations embody one of the fundamental concepts in algebra. They represent equations of the first degree, meaning the variables are not raised to a power higher than one. This creates a straight-line graph when plotted on a coordinate plane.

Here are the key components of a linear equation:

• They usually come in forms like slope-intercept (\(y = mx + b\)), point-slope (\(y - y_1 = m(x - x_1)\)), or standard form (\(Ax + By = C\)).
• The slope (\( m \)) in slope-intercept and point-slope forms defines the steepness or incline of the line.

Vertical lines are a bit of an exception. Their equations don't fit within these traditional forms because they have undefined slopes and are expressed as \( x = a \). Ultimately, what classifies them as 'linear' is not their slope or form, but the fact they graph as a line on a plane.

Understanding linear equations is key for solving many real-world problems involving relationships between two variables, like speed versus time graphs in physics. Even vertical lines, though unique in representation, fall under the grand umbrella of linear equations.