Problem 40 - 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 \((4,2)\) and perpendicular to the horizontal line passing through \((3,1)\)

Step-by-Step Solution

Answer

The equation of the line is \(x = 4\).

Step 1: Identify Line Characteristics

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

Step 2: Determine Equation of the Vertical Line

Since vertical lines are defined by constant x-values, and the line in question passes through (4, 2), the line's equation is simply \(x = 4\). This line is vertical because it does not change in the x-direction.

Step 3: Convert to Standard Form

For standard form, \(Ax + By = C\), vertical lines can be written as \(x = 4\), which in a more standard-like notation is \(x - 4 = 0\), corresponding to \(1 \cdot x + 0 \cdot y = 4\) or \(x + 0y = 4\).

Key Concepts

Vertical Lines

Vertical lines are an essential part of understanding line equations. Unlike horizontal lines, which run left to right, vertical lines go up and down. This means they do not have the typical slope, which measures a line's steepness. Instead, vertical lines have an undefined slope. When saying a line is vertical, it means it does not tilt from its vertical path. It moves straight up, parallel to the y-axis at all points.
Vertical lines are best understood through their constant x-value. No matter how far up or down you move along a vertical line, the x-coordinate does not change. For example, in the exercise where the line passes through the point (4, 2), every other point on this line will share the same x-coordinate, 4. Thus, the equation of the vertical line can easily be expressed simply as:

• x = constant

Remember, in vertical line equations:

• The line is parallel to the y-axis.
• The slope is undefined.

Standard Form

To express a line equation in standard form means arranging it as:\[ Ax + By = C \]where A, B, and C are integers, and A should ideally be a positive number. This form is valuable because it provides a consistent structure to analyze and compare different lines.
For vertical lines, because they primarily involve a constant x-value, the standard form looks slightly different but follows the general principle. For a vertical line such as the one from our exercise (where x = 4), writing it in standard form involves illustrating there is no y-component impact:

• 1\(x\) + 0\(y\) = 4

This translates to simply \( x - 4 = 0 \), where A = 1, B = 0, and C = 4. The zero coefficient on the y-variable shows there's no vertical (or y-directional) change in the line.

Slope

The slope of a line shows its steepness and direction, calculated as the ratio of rise over run between two points. When a line slants upwards, its slope is positive; if it slants downwards, the slope is negative.
However, the slope behaves differently for special types of lines like horizontal and vertical lines. While horizontal lines have a slope of 0, vertical lines have an undefined slope. This undefined slope arises because you cannot divide by zero, which happens when you attempt to calculate slope, as the run (change in x) for vertical lines is always 0.

• For horizontal lines: slope = 0
• For vertical lines: slope = undefined

Understandably, this difference in slope types reflects the distinct orientation of the lines:

• Horizontal lines: parallel to x-axis
• Vertical lines: parallel to y-axis