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 \((-1,-7)\)

Step-by-Step Solution

Answer

The equation of the line is \(y = 3\).

Step 1: Identify the Slope of the Vertical Line

A vertical line has an undefined slope. It is of the form \(x = c\), where \(c\) is a constant. The vertical line in question passes through \((-1, -7)\) and has the equation \(x = -1\).

Step 2: Determine the Slope of the Perpendicular Line

The slope of a line perpendicular to a vertical line is horizontal, which means the slope is \(0\). Therefore, the line perpendicular to \(x = -1\) is a horizontal line.

Step 3: Write the Equation of the Horizontal Line

A horizontal line has the equation \(y = b\), where \(b\) is the y-coordinate of any point on the line. Since this line passes through \((7,3)\), the equation is \(y = 3\).

Step 4: Put the Equation in Standard Form

The standard form of a line's equation is \(Ax + By = C\). For the horizontal line \(y = 3\), adjust it to standard form by making \(A = 0\), \(B = 1\), and \(C = 3\). Thus, the equation is currently \(0x + 1y = 3\), or simply \(y = 3\).

Key Concepts

Standard Form

Understanding the standard form of a line can be really helpful when dealing with different types of equations. The standard form of a line's equation is given by the format \( Ax + By = C \), where:

• \( A \), \( B \), and \( C \) are integers.
• \( A \) should not be negative.
• \( A \) and \( B \) should not both be zero.

This format is often used because it's a universal way to express linear equations. In our example, we had a horizontal line given by \( y = 3 \). To convert it to standard form, we see that \( A = 0 \), \( B = 1 \), and \( C = 3 \).
This conversion might seem trivial for a horizontal line, but it helps when consistent formatting is needed across different equations.

Perpendicular Lines

Perpendicular lines have distinct properties that define their relationship. When two lines are perpendicular, the product of their slopes is \(-1\).
However, there's a special case: when one of the lines is vertical, its slope is undefined. This means:

• A vertical line with the equation \( x = c \), where the slope is undefined.
• A line perpendicular to a vertical line is horizontal with a slope of \( 0 \).

In the exercise, the line we were given is perpendicular to the vertical line \( x = -1 \). This tells us the needed line is horizontal, with a slope of zero. This nature of perpendicular lines helps us quickly determine the required direction of our new line.

Horizontal Line

A horizontal line is unique in its own simple way. It is flat and runs parallel to the x-axis. The equation for a horizontal line is \( y = b \), where \( b \) is a constant.
This reflects the y-coordinate of any point the line goes through, meaning its vertical position stays the same as it stretches along the x-axis.
For instance, in our exercise, the line passes through the point \((7,3)\). This tells us the equation of the line is \( y = 3 \), indicating that anywhere you look on this line, the y-coordinate is \( 3 \).
Understanding horizontal lines can greatly simplify the process of graphing them and manipulating their equations, especially when combined with their standard form.