Problem 41 - 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 \((7,3)\) and perpendicular to the vertical line passing through \((-2,4)\)

Step-by-Step Solution

Answer

The standard form of the equation is \( 0x + 1y = 3 \).

Step 1: Understanding perpendicularity to a vertical line

A vertical line has an undefined slope and is of the form \( x = a \), where \( a \) is a constant. Any line perpendicular to a vertical line will be a horizontal line, which implies it has a slope of 0.

Step 2: Finding the slope of the requested line

Since the line needs to be perpendicular to a vertical line, it is a horizontal line. Therefore, the slope \( m \) is 0.

Step 3: Using the point-slope formula with a zero slope

The point-slope form of a line's equation is \( y - y_1 = m(x - x_1) \). For a slope of 0 and a point \((7, 3)\), the equation becomes \( y - 3 = 0(x - 7) \), which simplifies to \( y = 3 \).

Step 4: Converting to standard form

The standard form of a line equation is \( Ax + By = C \). Since our equation is \( y = 3 \), it can be rewritten in standard form as \( 0x + 1y = 3 \).

Key Concepts

Vertical and Horizontal Lines

Vertical and horizontal lines are vital concepts when learning about the properties and equations of lines on the coordinate plane.
A vertical line is characterized by having an equation in the form of \( x = a \), where \( a \) is the constant x-value for every point on the line. This means all points on a vertical line have the same x-coordinate, making it parallel to the y-axis. Since there is no change in the x-direction, the slope of a vertical line is undefined.
On the other hand, a horizontal line is defined by the equation \( y = b \), where \( b \) is the constant y-value for the line. This means that every point on the horizontal line shares the same y-coordinate, thereby being parallel to the x-axis. The slope of a horizontal line is 0, indicating no inclination to the left or right. When a line is perpendicular to a vertical line, it must be a horizontal line. This is because perpendicular lines intersect at right angles, and the only line that meets a vertical line at a right angle is a horizontal one.

Perpendicular Lines

Perpendicular lines intersect at right angles, which is a key concept in geometry.
When two lines are perpendicular, their slopes have a unique relationship. More specifically, the slopes are negative reciprocals of one another. That is, if one line has a slope of \( m \), the line perpendicular to it will have a slope of \(-\frac{1}{m}\).
However, when dealing with vertical and horizontal lines, this relationship adjusts since a vertical line has an undefined slope. The line perpendicular to a vertical line will always be a horizontal line, with a slope of 0. This exception is crucial in understanding the broader context of perpendicular lines.

Standard Form of a Line

The standard form of a line's equation is a common format that helps in analyzing line equations easily.
It is written as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should be non-negative.
Converting a line's equation into standard form involves rearranging terms to match this specific pattern. It's particularly useful when you need to determine the x- and y-intercepts quickly. In our exercise example, starting from the equation \( y = 3 \) (which represents a horizontal line), converting it to standard form is straightforward: \( 0x + 1y = 3 \). This conversion emphasizes that the line has no x-term, reinforcing its horizontal nature.