Problem 37 - 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 \((-2,3)\) and parallel to the vertical line passing through \((2,1)\)

Step-by-Step Solution

Answer

The equation of the line is \(x = -2\).

Step 1: Understand the Line Requirement

The line we are looking for needs to be parallel to a given line. The given line is a vertical line passing through \((2,1)\). All vertical lines have an undefined slope and can be represented by equations of the form \(x = a\), where \(a\) is a constant. Since this line passes through \((2,1)\), its equation is \(x = 2\).

Step 2: Determine the Equation of the Parallel Line

Since the sought line must be parallel to the given vertical line \(x = 2\), it too must be a vertical line. Parallel lines share the same slope, and as the original line is vertical, the new line will also have an undefined slope. That means its equation will also be of the form \(x = a\).

Step 3: Use the Given Point to Find 'a'

The equation of our desired line must pass through the point \((-2,3)\). For a vertical line, this means the x-coordinate of every point on the line, including \((-2,3)\), is the same as \(a\). Therefore, \(a = -2\).

Step 4: Write the Equation in Standard Form

For a vertical line, the standard form of the line equation is simple and remains \(x = -2\).

Key Concepts

Parallel Lines

Parallel lines are fascinating because of their special relationship. They travel in the same direction but never intersect. This means they maintain a constant distance apart and have identical slopes. In the context of line equations, if you have one line and want to find another line parallel to it, you'll need to maintain the slope. If the original line's equation is in the form \(y = mx + b\), then a parallel line would have an equation \(y = mx + c\), where \(m\) is the same slope and \(c\) is some y-intercept. A vital point to remember about parallel lines is that vertical lines also fit this category, but they represent a small twist. Vertical lines have an undefined slope, and their equations look like \(x = a\), showing they run up and down the graph, staying parallel to the y-axis.

Vertical Lines

Vertical lines are unique in the world of linear equations. They run straight up and down, aligning directly with the y-axis. Their property is that no matter how far up or down you travel, the x-coordinate remains the same for all points on the line. Because of this, the equation of a vertical line only involves \(x\) and takes the form \(x = a\), where \(a\) is a constant. This means all points on the line have an x-coordinate equal to \(a\). Unlike other lines, you won't find a 'y' in their equation, as the line covers all possible y-values. Vertical lines stand out because they have an undefined slope, which creates a special case when working with line equations.

Undefined Slope

The concept of undefined slope might sound puzzling at first, but it's a critical aspect of understanding some lines, particularly vertical ones. Slope, in simple terms, is a measure of how steep a line is. It's calculated as the change in y over the change in x (rise over run). When it comes to a vertical line, the 'run' or change in x is zero because every point on the line has the same x-coordinate. This creates a mathematical situation where you would have to divide by zero to find the slope, which is undefined. You might remember from math rules that division by zero is not possible, leading to the description of a vertical line's slope as undefined. This uniqueness adds an important distinction in line equations and helps illustrate why vertical lines don't conform to more typical slope-intercept equations.